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ThroughtThroughout this answer, I'll be referencing Kubert--Lang, "Modular Units" ChChapter 2, sections 1 and 2, and ChChapter 3 section 4.

Each of your functions is a ratio of the Siegel functions of the form $$A_N(\tau)=\left(\frac{g_{\frac1N,0}}{g_{\frac aN,0}}(N\tau)\right)^m.$$

If $r=(r_1,r_2)\in \mathbb Q^2$, $B_2(x)=x^2-x+1/6$ is the second Bernoulli polynomial, $q=e^{2\pi i \tau}$ as usual, and $\zeta=e^{2\pi i r_2}$, then the Siegel function $g_a$ is defined by $$ g_a(\tau)= -q^{\frac12 B_2(r_1)}e^{2\pi i \frac{r_2(r_1-1)}2}\prod_{n=0}^\infty(1-q^{n+r_1}\zeta)(1-q^{n+1-r_1}\zeta^{-1}). $$ The Siegel functions $g_a$ are modular units, meaning they're modular functions whose zeros and poles are supported at cusps.

If $\gamma\in \text{SL}_2(\mathbb Z)$, then $$ g_a|_0\gamma=g_{a\gamma}. $$ Moreover, we can reduce $r_1$ and $r_2$ modulo $\mathbb Z$, or change signs of both $r_1$ and $r_2$, if we introduce a root of unity. However, the $m$'s in your ratios satisfy $m(1-a^2)\equiv 0 \pmod N$ if $N$ is odd and $\pmod{2N}$ if $N$, and so using Theorem 3.4.1 of Kubert--Lang we find we can ignore this extra factor so the transformations rules becomes this simple. Note, the theorem is written in terms of Klein functions $t_a=g_a\Delta^{-\frac1{12}}$. Using these transformation rules, it's easy to see that all of your functions are invariant under $\Gamma_1(N)/\{\pm1\}$.

The specific groups you have considered are ones where this group is genus $0$, and your functions $A_N$ are HauptmodlnHauptmoduln with a unique zero at the cusp $\infty$, a constant $C_0$ at the cusp $0$, and a unique pole at one other cusp. We can calculate $C_0$ using the formulas above: If $\zeta_{2N}=e^{2\pi i \frac{1}{2N}}$, then the constant is given by $\left(\frac{\zeta_{2N}-\zeta_{2N}^{-1}}{\zeta_{2N}^{a}-\zeta_{2N}^{-a}}\right)^m$.

Since $A_N$ is a hauptmodl, its image under the Fricke involution is a rational function in $A_N$. Since it now has a unique $0$ at the cusp $0$ (Where $A_N$ has the constant $C_0$) and the constant $C_0$ at the cusp infinity (Where $A_N$ has a zero), we must have $$ A_N|_0\begin{pmatrix}0&-1\\N&0\end{pmatrix}=\frac{-A_N+C_0}{X\cdot A_N+1}, $$ for some $X$, determined by the movement of the pole. If the pole does not move, then $X=0$. Otherwise, $-1/X$ is the constant of $A_N$ at the cusp where $A_N|_0\begin{pmatrix}0&-1\\N&0\end{pmatrix}$ has its pole.

All in all, this suggest you might have hope for $N=7,9$ and $10$, which all give genus $0$ groups. Unfortunately The ratios of the type you've described above aren't hauptmodlnhauptmoduln. You might be able to find similar results if you allow for larger order products.

Throught this answer, I'll be referencing Kubert--Lang, "Modular Units" Ch 2, sections 1 and 2, and Ch 3 section 4.

Each of your functions is a ratio of the Siegel functions of the form $$A_N(\tau)=\left(\frac{g_{\frac1N,0}}{g_{\frac aN,0}}(N\tau)\right)^m.$$

If $r=(r_1,r_2)\in \mathbb Q^2$, $B_2(x)=x^2-x+1/6$ is the second Bernoulli polynomial, $q=e^{2\pi i \tau}$ as usual, and $\zeta=e^{2\pi i r_2}$, then the Siegel function $g_a$ is defined by $$ g_a(\tau)= -q^{\frac12 B_2(r_1)}e^{2\pi i \frac{r_2(r_1-1)}2}\prod_{n=0}^\infty(1-q^{n+r_1}\zeta)(1-q^{n+1-r_1}\zeta^{-1}). $$ The Siegel functions $g_a$ are modular units, meaning they're modular functions whose zeros and poles are supported at cusps.

If $\gamma\in \text{SL}_2(\mathbb Z)$, then $$ g_a|_0\gamma=g_{a\gamma}. $$ Moreover, we can reduce $r_1$ and $r_2$ modulo $\mathbb Z$, or change signs of both $r_1$ and $r_2$, if we introduce a root of unity. However, the $m$'s in your ratios satisfy $m(1-a^2)\equiv 0 \pmod N$ if $N$ is odd and $\pmod{2N}$ if $N$, and so using Theorem 3.4.1 of Kubert--Lang we find we can ignore this extra factor so the transformations rules becomes this simple. Note, the theorem is written in terms of Klein functions $t_a=g_a\Delta^{-\frac1{12}}$. Using these transformation rules, it's easy to see that all of your functions are invariant under $\Gamma_1(N)/\{\pm1\}$.

The specific groups you have considered are ones where this group is genus $0$, and your functions $A_N$ are Hauptmodln with a unique zero at the cusp $\infty$, a constant $C_0$ at the cusp $0$, and a unique pole at one other cusp. We can calculate $C_0$ using the formulas above: If $\zeta_{2N}=e^{2\pi i \frac{1}{2N}}$, then the constant is given by $\left(\frac{\zeta_{2N}-\zeta_{2N}^{-1}}{\zeta_{2N}^{a}-\zeta_{2N}^{-a}}\right)^m$.

Since $A_N$ is a hauptmodl, its image under the Fricke involution is a rational function in $A_N$. Since it now has a unique $0$ at the cusp $0$ (Where $A_N$ has the constant $C_0$) and the constant $C_0$ at the cusp infinity (Where $A_N$ has a zero), we must have $$ A_N|_0\begin{pmatrix}0&-1\\N&0\end{pmatrix}=\frac{-A_N+C_0}{X\cdot A_N+1}, $$ for some $X$, determined by the movement of the pole. If the pole does not move, then $X=0$. Otherwise, $-1/X$ is the constant of $A_N$ at the cusp where $A_N|_0\begin{pmatrix}0&-1\\N&0\end{pmatrix}$ has its pole.

All in all, this suggest you might have hope for $N=7,9$ and $10$, which all give genus $0$ groups. Unfortunately The ratios of the type you've described above aren't hauptmodln. You might be able to find similar results if you allow for larger order products.

Throughout this answer, I'll be referencing Kubert-Lang, "Modular Units" Chapter 2, sections 1 and 2, and Chapter 3 section 4.

Each of your functions is a ratio of the Siegel functions of the form $$A_N(\tau)=\left(\frac{g_{\frac1N,0}}{g_{\frac aN,0}}(N\tau)\right)^m.$$

If $r=(r_1,r_2)\in \mathbb Q^2$, $B_2(x)=x^2-x+1/6$ is the second Bernoulli polynomial, $q=e^{2\pi i \tau}$ as usual, and $\zeta=e^{2\pi i r_2}$, then the Siegel function $g_a$ is defined by $$ g_a(\tau)= -q^{\frac12 B_2(r_1)}e^{2\pi i \frac{r_2(r_1-1)}2}\prod_{n=0}^\infty(1-q^{n+r_1}\zeta)(1-q^{n+1-r_1}\zeta^{-1}). $$ The Siegel functions $g_a$ are modular units, meaning they're modular functions whose zeros and poles are supported at cusps.

If $\gamma\in \text{SL}_2(\mathbb Z)$, then $$ g_a|_0\gamma=g_{a\gamma}. $$ Moreover, we can reduce $r_1$ and $r_2$ modulo $\mathbb Z$, or change signs of both $r_1$ and $r_2$, if we introduce a root of unity. However, the $m$'s in your ratios satisfy $m(1-a^2)\equiv 0 \pmod N$ if $N$ is odd and $\pmod{2N}$ if $N$, and so using Theorem 3.4.1 of Kubert-Lang we find we can ignore this extra factor so the transformations rules becomes this simple. Note, the theorem is written in terms of Klein functions $t_a=g_a\Delta^{-\frac1{12}}$. Using these transformation rules, it's easy to see that all of your functions are invariant under $\Gamma_1(N)/\{\pm1\}$.

The specific groups you have considered are ones where this group is genus $0$, and your functions $A_N$ are Hauptmoduln with a unique zero at the cusp $\infty$, a constant $C_0$ at the cusp $0$, and a unique pole at one other cusp. We can calculate $C_0$ using the formulas above: If $\zeta_{2N}=e^{2\pi i \frac{1}{2N}}$, then the constant is given by $\left(\frac{\zeta_{2N}-\zeta_{2N}^{-1}}{\zeta_{2N}^{a}-\zeta_{2N}^{-a}}\right)^m$.

Since $A_N$ is a hauptmodl, its image under the Fricke involution is a rational function in $A_N$. Since it now has a unique $0$ at the cusp $0$ (Where $A_N$ has the constant $C_0$) and the constant $C_0$ at the cusp infinity (Where $A_N$ has a zero), we must have $$ A_N|_0\begin{pmatrix}0&-1\\N&0\end{pmatrix}=\frac{-A_N+C_0}{X\cdot A_N+1}, $$ for some $X$, determined by the movement of the pole. If the pole does not move, then $X=0$. Otherwise, $-1/X$ is the constant of $A_N$ at the cusp where $A_N|_0\begin{pmatrix}0&-1\\N&0\end{pmatrix}$ has its pole.

All in all, this suggest you might have hope for $N=7,9$ and $10$, which all give genus $0$ groups. Unfortunately The ratios of the type you've described above aren't hauptmoduln. You might be able to find similar results if you allow for larger order products.

Removed a bad example.
Source Link

Throught this answer, I'll be referencing Kubert--Lang, "Modular Units" Ch 2, sections 1 and 2, and Ch 3 section 4.

Each of your functions is a ratio of the Siegel functions of the form $$A_N(\tau)=\left(\frac{g_{\frac1N,0}}{g_{\frac aN,0}}(N\tau)\right)^m.$$

If $r=(r_1,r_2)\in \mathbb Q^2$, $B_2(x)=x^2-x+1/6$ is the second Bernoulli polynomial, $q=e^{2\pi i \tau}$ as usual, and $\zeta=e^{2\pi i r_2}$, then the Siegel function $g_a$ is defined by $$ g_a(\tau)= -q^{\frac12 B_2(r_1)}e^{2\pi i \frac{r_2(r_1-1)}2}\prod_{n=0}^\infty(1-q^{n+r_1}\zeta)(1-q^{n+1-r_1}\zeta^{-1}). $$ The Siegel functions $g_a$ are modular units, meaning they're modular functions whose zeros and poles are supported at cusps.

If $\gamma\in \text{SL}_2(\mathbb Z)$, then $$ g_a|_0\gamma=g_{a\gamma}. $$ Moreover, we can reduce $r_1$ and $r_2$ modulo $\mathbb Z$, or change signs of both $r_1$ and $r_2$, if we introduce a root of unity. However, the $m$'s in your ratios satisfy $m(1-a^2)\equiv 0 \pmod N$ if $N$ is odd and $\pmod{2N}$ if $N$, and so using Theorem 3.4.1 of Kubert--Lang we find we can ignore this extra factor so the transformations rules becomes this simple. Note, the theorem is written in terms of Klein functions $t_a=g_a\Delta^{-\frac1{12}}$. Using these transformation rules, it's easy to see that all of your functions are invariant under $\Gamma_1(N)/\{\pm1\}$.

The specific groups you have considered are ones where this group is genus $0$, and your functions $A_N$ are Hauptmodln with a unique zero at the cusp $\infty$, a constant $C_0$ at the cusp $0$, and a unique pole at one other cusp. We can calculate $C_0$ using the formulas above: If $\zeta_{2N}=e^{2\pi i \frac{1}{2N}}$, then the constant is given by $\left(\frac{\zeta_{2N}-\zeta_{2N}^{-1}}{\zeta_{2N}^{a}-\zeta_{2N}^{-a}}\right)^m$.

Since $A_N$ is a hauptmodl, its image under the Fricke involution is a rational function in $A_N$. Since it now has a unique $0$ at the cusp $0$ (Where $A_N$ has the constant $C_0$) and the constant $C_0$ at the cusp infinity (Where $A_N$ has a zero), we must have $$ A_N|_0\begin{pmatrix}0&-1\\N&0\end{pmatrix}=\frac{-A_N+C_0}{X\cdot A_N+1}, $$ for some $X$, determined by the movement of the pole. If the pole does not move, then $X=0$. Otherwise, $-1/X$ is the constant of $A_N$ at the cusp where $A_N|_0\begin{pmatrix}0&-1\\N&0\end{pmatrix}$ has its pole.

All in all, this suggest you might have hope for $N=7,9$ and $10$, which all give genus $0$ groups. Unfortunately The ratios of the type you've described above aren't hauptmodln. You might be able to find similar results if you allow for larger order products. For instance, if $P\equiv 1\pmod 4$ is prime and $\chi$ is the quadratic character mod $P$, then the product $$ B_P(\tau):=\prod_{a=1}^{\frac{P-1}2} g_{\frac aP,0}(p\tau)^{\chi(a)} $$ should satisfy $$\left(B_P\right)^m|_0\begin{pmatrix}0&-1\\P&0\end{pmatrix}=C_0-\left(B_P\right)^m$$ for appropriate $m$, since this is modular on a group with three distinct cusp: The cusp $\infty$ where the function has all its zeros, the cusp $0$, where the function has a constant, and one other cusp fixed by the Fricke involution, where the expansion is given by
$$\left(B_P\right)^m|_0\begin{pmatrix}a&b\\c&d\end{pmatrix}=\pm \left(B_P\right)^{-m},$$ where $c\equiv 0\pmod P$ and $\chi(d)=-1$.

Throught this answer, I'll be referencing Kubert--Lang, "Modular Units" Ch 2, sections 1 and 2, and Ch 3 section 4.

Each of your functions is a ratio of the Siegel functions of the form $$A_N(\tau)=\left(\frac{g_{\frac1N,0}}{g_{\frac aN,0}}(N\tau)\right)^m.$$

If $r=(r_1,r_2)\in \mathbb Q^2$, $B_2(x)=x^2-x+1/6$ is the second Bernoulli polynomial, $q=e^{2\pi i \tau}$ as usual, and $\zeta=e^{2\pi i r_2}$, then the Siegel function $g_a$ is defined by $$ g_a(\tau)= -q^{\frac12 B_2(r_1)}e^{2\pi i \frac{r_2(r_1-1)}2}\prod_{n=0}^\infty(1-q^{n+r_1}\zeta)(1-q^{n+1-r_1}\zeta^{-1}). $$ The Siegel functions $g_a$ are modular units, meaning they're modular functions whose zeros and poles are supported at cusps.

If $\gamma\in \text{SL}_2(\mathbb Z)$, then $$ g_a|_0\gamma=g_{a\gamma}. $$ Moreover, we can reduce $r_1$ and $r_2$ modulo $\mathbb Z$, or change signs of both $r_1$ and $r_2$, if we introduce a root of unity. However, the $m$'s in your ratios satisfy $m(1-a^2)\equiv 0 \pmod N$ if $N$ is odd and $\pmod{2N}$ if $N$, and so using Theorem 3.4.1 of Kubert--Lang we find we can ignore this extra factor so the transformations rules becomes this simple. Note, the theorem is written in terms of Klein functions $t_a=g_a\Delta^{-\frac1{12}}$. Using these transformation rules, it's easy to see that all of your functions are invariant under $\Gamma_1(N)/\{\pm1\}$.

The specific groups you have considered are ones where this group is genus $0$, and your functions $A_N$ are Hauptmodln with a unique zero at the cusp $\infty$, a constant $C_0$ at the cusp $0$, and a unique pole at one other cusp. We can calculate $C_0$ using the formulas above: If $\zeta_{2N}=e^{2\pi i \frac{1}{2N}}$, then the constant is given by $\left(\frac{\zeta_{2N}-\zeta_{2N}^{-1}}{\zeta_{2N}^{a}-\zeta_{2N}^{-a}}\right)^m$.

Since $A_N$ is a hauptmodl, its image under the Fricke involution is a rational function in $A_N$. Since it now has a unique $0$ at the cusp $0$ (Where $A_N$ has the constant $C_0$) and the constant $C_0$ at the cusp infinity (Where $A_N$ has a zero), we must have $$ A_N|_0\begin{pmatrix}0&-1\\N&0\end{pmatrix}=\frac{-A_N+C_0}{X\cdot A_N+1}, $$ for some $X$, determined by the movement of the pole. If the pole does not move, then $X=0$. Otherwise, $-1/X$ is the constant of $A_N$ at the cusp where $A_N|_0\begin{pmatrix}0&-1\\N&0\end{pmatrix}$ has its pole.

All in all, this suggest you might have hope for $N=7,9$ and $10$, which all give genus $0$ groups. Unfortunately The ratios of the type you've described above aren't hauptmodln. You might be able to find similar results if you allow for larger order products. For instance, if $P\equiv 1\pmod 4$ is prime and $\chi$ is the quadratic character mod $P$, then the product $$ B_P(\tau):=\prod_{a=1}^{\frac{P-1}2} g_{\frac aP,0}(p\tau)^{\chi(a)} $$ should satisfy $$\left(B_P\right)^m|_0\begin{pmatrix}0&-1\\P&0\end{pmatrix}=C_0-\left(B_P\right)^m$$ for appropriate $m$, since this is modular on a group with three distinct cusp: The cusp $\infty$ where the function has all its zeros, the cusp $0$, where the function has a constant, and one other cusp fixed by the Fricke involution, where the expansion is given by
$$\left(B_P\right)^m|_0\begin{pmatrix}a&b\\c&d\end{pmatrix}=\pm \left(B_P\right)^{-m},$$ where $c\equiv 0\pmod P$ and $\chi(d)=-1$.

Throught this answer, I'll be referencing Kubert--Lang, "Modular Units" Ch 2, sections 1 and 2, and Ch 3 section 4.

Each of your functions is a ratio of the Siegel functions of the form $$A_N(\tau)=\left(\frac{g_{\frac1N,0}}{g_{\frac aN,0}}(N\tau)\right)^m.$$

If $r=(r_1,r_2)\in \mathbb Q^2$, $B_2(x)=x^2-x+1/6$ is the second Bernoulli polynomial, $q=e^{2\pi i \tau}$ as usual, and $\zeta=e^{2\pi i r_2}$, then the Siegel function $g_a$ is defined by $$ g_a(\tau)= -q^{\frac12 B_2(r_1)}e^{2\pi i \frac{r_2(r_1-1)}2}\prod_{n=0}^\infty(1-q^{n+r_1}\zeta)(1-q^{n+1-r_1}\zeta^{-1}). $$ The Siegel functions $g_a$ are modular units, meaning they're modular functions whose zeros and poles are supported at cusps.

If $\gamma\in \text{SL}_2(\mathbb Z)$, then $$ g_a|_0\gamma=g_{a\gamma}. $$ Moreover, we can reduce $r_1$ and $r_2$ modulo $\mathbb Z$, or change signs of both $r_1$ and $r_2$, if we introduce a root of unity. However, the $m$'s in your ratios satisfy $m(1-a^2)\equiv 0 \pmod N$ if $N$ is odd and $\pmod{2N}$ if $N$, and so using Theorem 3.4.1 of Kubert--Lang we find we can ignore this extra factor so the transformations rules becomes this simple. Note, the theorem is written in terms of Klein functions $t_a=g_a\Delta^{-\frac1{12}}$. Using these transformation rules, it's easy to see that all of your functions are invariant under $\Gamma_1(N)/\{\pm1\}$.

The specific groups you have considered are ones where this group is genus $0$, and your functions $A_N$ are Hauptmodln with a unique zero at the cusp $\infty$, a constant $C_0$ at the cusp $0$, and a unique pole at one other cusp. We can calculate $C_0$ using the formulas above: If $\zeta_{2N}=e^{2\pi i \frac{1}{2N}}$, then the constant is given by $\left(\frac{\zeta_{2N}-\zeta_{2N}^{-1}}{\zeta_{2N}^{a}-\zeta_{2N}^{-a}}\right)^m$.

Since $A_N$ is a hauptmodl, its image under the Fricke involution is a rational function in $A_N$. Since it now has a unique $0$ at the cusp $0$ (Where $A_N$ has the constant $C_0$) and the constant $C_0$ at the cusp infinity (Where $A_N$ has a zero), we must have $$ A_N|_0\begin{pmatrix}0&-1\\N&0\end{pmatrix}=\frac{-A_N+C_0}{X\cdot A_N+1}, $$ for some $X$, determined by the movement of the pole. If the pole does not move, then $X=0$. Otherwise, $-1/X$ is the constant of $A_N$ at the cusp where $A_N|_0\begin{pmatrix}0&-1\\N&0\end{pmatrix}$ has its pole.

All in all, this suggest you might have hope for $N=7,9$ and $10$, which all give genus $0$ groups. Unfortunately The ratios of the type you've described above aren't hauptmodln. You might be able to find similar results if you allow for larger order products.

Edited to fix errors.
Source Link

Throught this answer, I'll be referencing Kubert--Lang, "Modular Units" Ch 2, sections 1 and 2, and Ch 3 section 4.

Each of your functions is a ratio of the Siegel functions of the form $$A_N(\tau)=\left(\frac{g_{\frac1N,0}}{g_{\frac aN,0}}(N\tau)\right)^m.$$

If $r=(r_1,r_2)\in \mathbb Q^2$, $B_2(x)=x^2-x+1/6$ is the second Bernoulli polynomial, $q=e^{2\pi i \tau}$ as usual, and $\zeta=e^{2\pi i r_2}$, then the Siegel function $g_a$ is defined by $$ g_a(\tau)= -q^{\frac12 B_2(r_1)}e^{2\pi i \frac{r_2(r_1-1)}2}\prod_{n=0}^\infty(1-q^{n+r_1}\zeta)(1-q^{n+1-r_1}\zeta^{-1}). $$ The Siegel functions $g_a$ are modular units, meaning they're modular functions whose zeros and poles are supported at cusps.

If $\gamma\in \text{SL}_2(\mathbb Z)$, then $$ g_a|_0\gamma=g_{a\gamma}. $$ Moreover, we can reduce $r_1$ and $r_2$ modulo $\mathbb Z$, or change signs of both $r_1$ and $r_2$, if we introduce a root of unity. However, the $m$'s in your ratios satisfy $m(1-a^2)\equiv 0 \pmod N$ if $N$ is odd and $\pmod{2N}$ if $N$, and so using Theorem 3.4.1 of Kubert--Lang we find we can ignore this extra factor so the transformations rules becomes this simple. Note, the theorem is written in terms of Klein functions $t_a=g_a\Delta^{-\frac1{12}}$. Using these transformation rules, it's easy to see that all of your functions are invariant under $\Gamma_1(N)/\{\pm1\}$.

The specific groups you have considered are ones where this group is genus $0$, and your functions $A_N$ are Hauptmodln with a unique zero at the cusp $\infty$, a constant $C_0$ at the cusp $0$, and a unique pole at one other cusp. We can calculate $C_0$ using the formulas above: If $\zeta_{2N}=e^{2\pi i \frac{1}{2N}}$, then the constant is given by $\frac{\zeta_{2N}-\zeta_{2N}^{-1}}{\zeta_{2N}^{a}-\zeta_{2N}^{-a}}$$\left(\frac{\zeta_{2N}-\zeta_{2N}^{-1}}{\zeta_{2N}^{a}-\zeta_{2N}^{-a}}\right)^m$.

Since $A_N$ is a hauptmodl, its image under the Fricke involution is a rational function in $A_N$. Since it now has a unique $0$ at the cusp $0$ (Where $A_N$ has the constant $C_0$) and the constant $C_0$ at the cusp infinity (Where $A_N$ has a zero), we must have $$ A_N|_0\begin{pmatrix}0&-1\\N&0\end{pmatrix}=\frac{-A_N+C_0}{X\cdot A_N+1}, $$ for some $X$, determined by the movement of the pole. If the pole does not move, then $X=0$. Otherwise, $1/X$$-1/X$ is the constant of $A_N$ at the cusp where $A_N|_0\begin{pmatrix}0&-1\\N&0\end{pmatrix}$ has its pole.

All in all, this suggest you might have hope for $N=7,9$ and $10$, which all give genus $0$ groups. Unfortunately The ratios of the type you've described above aren't hauptmodln. You might be able to find similar results if you allow for larger order products. For instance, if $P\equiv 1\pmod 4$ is prime and $\chi$ is the quadratic character mod $P$, then the product $$ B_P(\tau):=\prod_{a=1}^{\frac{P-1}2} g_{\frac aP,0}(p\tau)^{\chi(a)} $$ should satisfy $$\left(B_P\right)^m|_0\begin{pmatrix}0&-1\\P&0\end{pmatrix}=C_0-\left(B_P\right)^m$$ for appropriate $m$, since this is modular on a group with three distinct cusp: The cusp $\infty$ where the function has all its zeros, the cusp $0$, where the function has a constant, and one other cusp fixed by the Fricke involution, where the expansion is given by
$$\left(B_P\right)^m|_0\begin{pmatrix}a&b\\c&d\end{pmatrix}=\pm \left(B_P\right)^{-m},$$ where $c\equiv 0\pmod P$ and $\chi(d)=-1$.

Throught this answer, I'll be referencing Kubert--Lang, "Modular Units" Ch 2, sections 1 and 2, and Ch 3 section 4.

Each of your functions is a ratio of the Siegel functions of the form $$A_N(\tau)=\left(\frac{g_{\frac1N,0}}{g_{\frac aN,0}}(N\tau)\right)^m.$$

If $r=(r_1,r_2)\in \mathbb Q^2$, $B_2(x)=x^2-x+1/6$ is the second Bernoulli polynomial, $q=e^{2\pi i \tau}$ as usual, and $\zeta=e^{2\pi i r_2}$, then the Siegel function $g_a$ is defined by $$ g_a(\tau)= -q^{\frac12 B_2(r_1)}e^{2\pi i \frac{r_2(r_1-1)}2}\prod_{n=0}^\infty(1-q^{n+r_1}\zeta)(1-q^{n+1-r_1}\zeta^{-1}). $$ The Siegel functions $g_a$ are modular units, meaning they're modular functions whose zeros and poles are supported at cusps.

If $\gamma\in \text{SL}_2(\mathbb Z)$, then $$ g_a|_0\gamma=g_{a\gamma}. $$ Moreover, we can reduce $r_1$ and $r_2$ modulo $\mathbb Z$, or change signs of both $r_1$ and $r_2$, if we introduce a root of unity. However, the $m$'s in your ratios satisfy $m(1-a^2)\equiv 0 \pmod N$ if $N$ is odd and $\pmod{2N}$ if $N$, and so using Theorem 3.4.1 of Kubert--Lang we find we can ignore this extra factor so the transformations rules becomes this simple. Note, the theorem is written in terms of Klein functions $t_a=g_a\Delta^{-\frac1{12}}$. Using these transformation rules, it's easy to see that all of your functions are invariant under $\Gamma_1(N)/\{\pm1\}$.

The specific groups you have considered are ones where this group is genus $0$, and your functions $A_N$ are Hauptmodln with a unique zero at the cusp $\infty$, a constant $C_0$ at the cusp $0$, and a unique pole at one other cusp. We can calculate $C_0$ using the formulas above: If $\zeta_{2N}=e^{2\pi i \frac{1}{2N}}$, then the constant is given by $\frac{\zeta_{2N}-\zeta_{2N}^{-1}}{\zeta_{2N}^{a}-\zeta_{2N}^{-a}}$.

Since $A_N$ is a hauptmodl, its image under the Fricke involution is a rational function in $A_N$. Since it now has a unique $0$ at the cusp $0$ (Where $A_N$ has the constant $C_0$) and the constant $C_0$ at the cusp infinity (Where $A_N$ has a zero), we must have $$ A_N|_0\begin{pmatrix}0&-1\\N&0\end{pmatrix}=\frac{-A_N+C_0}{X\cdot A_N+1}, $$ for some $X$, determined by the movement of the pole. If the pole does not move, then $X=0$. Otherwise, $1/X$ is the constant of $A_N$ at the cusp where $A_N|_0\begin{pmatrix}0&-1\\N&0\end{pmatrix}$ has its pole.

All in all, this suggest you might have hope for $N=7,9$ and $10$, which all give genus $0$ groups. Unfortunately The ratios of the type you've described above aren't hauptmodln. You might be able to find similar results if you allow for larger order products. For instance, if $P\equiv 1\pmod 4$ is prime and $\chi$ is the quadratic character mod $P$, then the product $$ B_P(\tau):=\prod_{a=1}^{\frac{P-1}2} g_{\frac aP,0}(p\tau)^{\chi(a)} $$ should satisfy $$\left(B_P\right)^m|_0\begin{pmatrix}0&-1\\P&0\end{pmatrix}=C_0-\left(B_P\right)^m$$ for appropriate $m$, since this is modular on a group with three distinct cusp: $\infty$ where the function has all its zeros, $0$, where the function has a constant, and one other cusp fixed by the Fricke involution, where the expansion is given by
$$\left(B_P\right)^m|_0\begin{pmatrix}a&b\\c&d\end{pmatrix}=\pm \left(B_P\right)^{-m},$$ where $c\equiv 0\pmod P$ and $\chi(d)=-1$.

Throught this answer, I'll be referencing Kubert--Lang, "Modular Units" Ch 2, sections 1 and 2, and Ch 3 section 4.

Each of your functions is a ratio of the Siegel functions of the form $$A_N(\tau)=\left(\frac{g_{\frac1N,0}}{g_{\frac aN,0}}(N\tau)\right)^m.$$

If $r=(r_1,r_2)\in \mathbb Q^2$, $B_2(x)=x^2-x+1/6$ is the second Bernoulli polynomial, $q=e^{2\pi i \tau}$ as usual, and $\zeta=e^{2\pi i r_2}$, then the Siegel function $g_a$ is defined by $$ g_a(\tau)= -q^{\frac12 B_2(r_1)}e^{2\pi i \frac{r_2(r_1-1)}2}\prod_{n=0}^\infty(1-q^{n+r_1}\zeta)(1-q^{n+1-r_1}\zeta^{-1}). $$ The Siegel functions $g_a$ are modular units, meaning they're modular functions whose zeros and poles are supported at cusps.

If $\gamma\in \text{SL}_2(\mathbb Z)$, then $$ g_a|_0\gamma=g_{a\gamma}. $$ Moreover, we can reduce $r_1$ and $r_2$ modulo $\mathbb Z$, or change signs of both $r_1$ and $r_2$, if we introduce a root of unity. However, the $m$'s in your ratios satisfy $m(1-a^2)\equiv 0 \pmod N$ if $N$ is odd and $\pmod{2N}$ if $N$, and so using Theorem 3.4.1 of Kubert--Lang we find we can ignore this extra factor so the transformations rules becomes this simple. Note, the theorem is written in terms of Klein functions $t_a=g_a\Delta^{-\frac1{12}}$. Using these transformation rules, it's easy to see that all of your functions are invariant under $\Gamma_1(N)/\{\pm1\}$.

The specific groups you have considered are ones where this group is genus $0$, and your functions $A_N$ are Hauptmodln with a unique zero at the cusp $\infty$, a constant $C_0$ at the cusp $0$, and a unique pole at one other cusp. We can calculate $C_0$ using the formulas above: If $\zeta_{2N}=e^{2\pi i \frac{1}{2N}}$, then the constant is given by $\left(\frac{\zeta_{2N}-\zeta_{2N}^{-1}}{\zeta_{2N}^{a}-\zeta_{2N}^{-a}}\right)^m$.

Since $A_N$ is a hauptmodl, its image under the Fricke involution is a rational function in $A_N$. Since it now has a unique $0$ at the cusp $0$ (Where $A_N$ has the constant $C_0$) and the constant $C_0$ at the cusp infinity (Where $A_N$ has a zero), we must have $$ A_N|_0\begin{pmatrix}0&-1\\N&0\end{pmatrix}=\frac{-A_N+C_0}{X\cdot A_N+1}, $$ for some $X$, determined by the movement of the pole. If the pole does not move, then $X=0$. Otherwise, $-1/X$ is the constant of $A_N$ at the cusp where $A_N|_0\begin{pmatrix}0&-1\\N&0\end{pmatrix}$ has its pole.

All in all, this suggest you might have hope for $N=7,9$ and $10$, which all give genus $0$ groups. Unfortunately The ratios of the type you've described above aren't hauptmodln. You might be able to find similar results if you allow for larger order products. For instance, if $P\equiv 1\pmod 4$ is prime and $\chi$ is the quadratic character mod $P$, then the product $$ B_P(\tau):=\prod_{a=1}^{\frac{P-1}2} g_{\frac aP,0}(p\tau)^{\chi(a)} $$ should satisfy $$\left(B_P\right)^m|_0\begin{pmatrix}0&-1\\P&0\end{pmatrix}=C_0-\left(B_P\right)^m$$ for appropriate $m$, since this is modular on a group with three distinct cusp: The cusp $\infty$ where the function has all its zeros, the cusp $0$, where the function has a constant, and one other cusp fixed by the Fricke involution, where the expansion is given by
$$\left(B_P\right)^m|_0\begin{pmatrix}a&b\\c&d\end{pmatrix}=\pm \left(B_P\right)^{-m},$$ where $c\equiv 0\pmod P$ and $\chi(d)=-1$.

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