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I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2,

$$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, \eta^2(4\tau)\,\sqrt[3]{j(4\tau)}$$

which has the nice q-expansion (A027652, negated terms),

$$g(\tau) = 1/q - 2 + 248q^3 - 492q^4 +(15^3+744)q^7 + \dots + (5280^3+744)q^{67} + \dots + (640320^3+744)q^{163}+\dots$$

However, one can use other Eisenstein series $E_k(\tau)$ as the one below.

II. In a paper by Bruinier (p.6), Borcherds defines a modular form of weight 1/2. First let,

$$K(\tau) = \tfrac{1}{16}\big(\vartheta_3^4(\tau) - \tfrac{1}{8}\vartheta_2^4(\tau)\big)\,\vartheta_2^4(\tau)\vartheta_3(\tau)\vartheta_4^4(\tau)$$

then,

$$b(\tau) = 60\vartheta_3(\tau)+\frac{K(\tau)E_6(4\tau)}{\eta^{24}(4\tau)} = 60\vartheta_3(\tau)+\frac{K(\tau)\sqrt{j(4\tau)-1728}}{\eta^{12}(4\tau)} $$

This has the q-expansion (A013953),

$$b(\tau) = 1/q^3 + 4 - 240q + (\color{blue}{27000}-240)q^4 - \color{blue}{85995}q^5 + \color{blue}{1707264}q^8 - (\color{blue}{4096000}+240)q^9 + \color{blue}{44330496}q^{12}-91951146q^{13}+\dots$$

I noticed that the blue numbers appear in the degrees of irreducible representations of the Thompson group $Th$, given by the finite sequence of 48 integers (A003916),

$$1, 248, 4123, \color{blue}{27000}, 30628, 30875, 61256, \color{blue}{85995}, 147250, 767637, 779247, 957125, \color{blue}{1707264}, 2450240, 2572752, 3376737, \color{blue}{4096000}, 4123000, 4881384, 4936750,\dots\color{blue}{44330496},\dots 91171899, 111321000, 190373976.$$

with repeated terms deleted for brevity.

Questions:

1. If it is not coincidence, what is the reason?
2. In Bruinier (p.6) he says one can use $E_k(\tau)$ for $k=4,6,8,10,14$. I know $E_4(\tau)$ appears in a function that connects to the Baby Monster $F_2$. Going out on a limb, maybe $E_6(\tau)$ is for the Thompson group $F_3$ (as above), then $E_{10}(\tau)$ for the Harada-Norton $F_5$, and $E_{14}(\tau)$ for the Held group $F_7$?

I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2,

$$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, \eta^2(4\tau)\,\sqrt[3]{j(4\tau)}$$

which has the nice q-expansion (A027652, negated terms),

$$g(\tau) = 1/q - 2 + 248q^3 - 492q^4 +(15^3+744)q^7 + \dots + (5280^3+744)q^{67} + \dots + (640320^3+744)q^{163}+\dots$$

However, one can use other Eisenstein series $E_k(\tau)$ as the one below.

II. In a paper by Bruinier (p.6), Borcherds defines a modular form of weight 1/2. First let,

$$K(\tau) = \tfrac{1}{16}\big(\vartheta_3^4(\tau) - \tfrac{1}{8}\vartheta_2^4(\tau)\big)\,\vartheta_2^4(\tau)\vartheta_3(\tau)\vartheta_4^4(\tau)$$

then,

$$b(\tau) = 60\vartheta_3(\tau)+\frac{K(\tau)E_6(4\tau)}{\eta^{24}(4\tau)} = 60\vartheta_3(\tau)+\frac{K(\tau)\sqrt{j(4\tau)-1728}}{\eta^{12}(4\tau)} $$

This has the q-expansion (A013953),

$$b(\tau) = 1/q^3 + 4 - 240q + (\color{blue}{27000}-240)q^4 - \color{blue}{85995}q^5 + \color{blue}{1707264}q^8 - (\color{blue}{4096000}+240)q^9 + \color{blue}{44330496}q^{12}-91951146q^{13}+\dots$$

I noticed that the blue numbers appear in the degrees of irreducible representations of the Thompson group $Th$, given by the finite sequence of 48 integers (A003916),

$$1, 248, 4123, \color{blue}{27000}, 30628, 30875, 61256, \color{blue}{85995}, 147250, 767637, 779247, 957125, \color{blue}{1707264}, 2450240, 2572752, 3376737, \color{blue}{4096000}, 4123000, 4881384, 4936750,\dots\color{blue}{44330496},\dots 91171899, 111321000, 190373976.$$

with repeated terms deleted for brevity.

Questions:

1. If it is not coincidence, what is the reason?
2. In Bruinier (p.6) he says one can use $E_k(\tau)$ for $k=4,6,8,10,14$. What are the other functions for higher $k$?

Edit: As pointed out by S. Carnahan below, there is already a known moonshine for $Th$. A partial(?) list of others from Griess' "Happy Family" can be found in Monstrous Moonshine, such as for the Higman-Sims $HS$ and so on. (Are there more?)

I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2,

$$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, \eta^2(4\tau)\,\sqrt[3]{j(4\tau)}$$

which has the nice q-expansion (A027652, negated terms),

$$g(\tau) = 1/q - 2 + 248q^3 - 492q^4 +(15^3+744)q^7 + \dots + (5280^3+744)q^{67} + \dots + (640320^3+744)q^{163}+\dots$$

However, one can use other Eisenstein series $E_n(\tau)$ as the one below.

II. In a paper by Bruinier (p.6), Borcherds defines a modular form of weight 1/2. First let,

$$K(\tau) = \tfrac{1}{16}\big(\vartheta_3^4(\tau) - \tfrac{1}{8}\vartheta_2^4(\tau)\big)\,\vartheta_2^4(\tau)\vartheta_3(\tau)\vartheta_4^4(\tau)$$

then,

$$b(\tau) = 60\vartheta_3(\tau)+\frac{K(\tau)E_6(4\tau)}{\eta^{24}(4\tau)} = 60\vartheta_3(\tau)+\frac{K(\tau)\sqrt{j(4\tau)-1728}}{\eta^{12}(4\tau)} $$

This has the q-expansion (A013953),

$$b(\tau) = 1/q^3 + 4 - 240q + (\color{blue}{27000}-240)q^4 - \color{blue}{85995}q^5 + \color{blue}{1707264}q^8 - (\color{blue}{4096000}+240)q^9 + \color{blue}{44330496}q^{12}-91951146q^{13}+\dots$$

I noticed that the blue numbers appear in the degrees of irreducible representations of the Thompson group $Th$, given by the finite sequence of 48 integers (A003916),

$$1, 248, 4123, \color{blue}{27000}, 30628, 30875, 61256, \color{blue}{85995}, 147250, 767637, 779247, 957125, \color{blue}{1707264}, 2450240, 2572752, 3376737, \color{blue}{4096000}, 4123000, 4881384, 4936750,\dots\color{blue}{44330496},\dots 91171899, 111321000, 190373976.$$

with repeated terms deleted for brevity.

Questions:

1. If it is not coincidence, what is the reason?
2. If we use other $E_n(\tau)$, would there be "connections" to other sporadic groups?

I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2,

$$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, \eta^2(4\tau)\,\sqrt[3]{j(4\tau)}$$

which has the nice q-expansion (A027652, negated terms),

$$g(\tau) = 1/q - 2 + 248q^3 - 492q^4 +(15^3+744)q^7 + \dots + (5280^3+744)q^{67} + \dots + (640320^3+744)q^{163}+\dots$$

However, one can use other Eisenstein series $E_k(\tau)$ as the one below.

II. In a paper by Bruinier (p.6), Borcherds defines a modular form of weight 1/2. First let,

$$K(\tau) = \tfrac{1}{16}\big(\vartheta_3^4(\tau) - \tfrac{1}{8}\vartheta_2^4(\tau)\big)\,\vartheta_2^4(\tau)\vartheta_3(\tau)\vartheta_4^4(\tau)$$

then,

$$b(\tau) = 60\vartheta_3(\tau)+\frac{K(\tau)E_6(4\tau)}{\eta^{24}(4\tau)} = 60\vartheta_3(\tau)+\frac{K(\tau)\sqrt{j(4\tau)-1728}}{\eta^{12}(4\tau)} $$

This has the q-expansion (A013953),

$$b(\tau) = 1/q^3 + 4 - 240q + (\color{blue}{27000}-240)q^4 - \color{blue}{85995}q^5 + \color{blue}{1707264}q^8 - (\color{blue}{4096000}+240)q^9 + \color{blue}{44330496}q^{12}-91951146q^{13}+\dots$$

I noticed that the blue numbers appear in the degrees of irreducible representations of the Thompson group $Th$, given by the finite sequence of 48 integers (A003916),

$$1, 248, 4123, \color{blue}{27000}, 30628, 30875, 61256, \color{blue}{85995}, 147250, 767637, 779247, 957125, \color{blue}{1707264}, 2450240, 2572752, 3376737, \color{blue}{4096000}, 4123000, 4881384, 4936750,\dots\color{blue}{44330496},\dots 91171899, 111321000, 190373976.$$

with repeated terms deleted for brevity.

Questions:

1. If it is not coincidence, what is the reason?
2. In Bruinier (p.6) he says one can use $E_k(\tau)$ for $k=4,6,8,10,14$. I know $E_4(\tau)$ appears in a function that connects to the Baby Monster $F_2$. Going out on a limb, maybe $E_6(\tau)$ is for the Thompson group $F_3$ (as above), then $E_{10}(\tau)$ for the Harada-Norton $F_5$, and $E_{14}(\tau)$ for the Held group $F_7$?

I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2,

$$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, \eta^2(4\tau)\,\sqrt[3]{j(4\tau)}$$

which has the nice q-expansion (A027652, negated terms),

$$g(\tau) = 1/q - 2 + 248q^3 - 492q^4 +(15^3+744)q^7 + \dots + (5280^3+744)q^{67} + \dots + (640320^3+744)q^{163}+\dots$$

However, one can use other Eisenstein series $E_n(\tau)$ as the one below.

II. In a paper by Bruinier (p.6), Borcherds defines a modular form of weight 1/2. First let,

$$K(\tau) = \tfrac{1}{16}\big(\vartheta_3^4(\tau) - \tfrac{1}{8}\vartheta_2^4(\tau)\big)\,\vartheta_2^4(\tau)\vartheta_3(\tau)\vartheta_4^4(\tau)$$

then,

$$b(\tau) = 60\vartheta_3(\tau)+\frac{K(\tau)E_6(4\tau)}{\eta^{24}(4\tau)} = 60\vartheta_3(\tau)+\frac{K(\tau)\sqrt{j(4\tau)-1728}}{\eta^{12}(4\tau)} $$

This has the q-expansion (A013953),

$$b(\tau) = 1/q^3 + 4 - 240q + (\color{blue}{27000}-240)q^4 - \color{blue}{85995}q^5 + \color{blue}{1707264}q^8 - (\color{blue}{4096000}+240)q^9 + \color{blue}{44330496}q^{12}-91951146q^{13}+\dots$$

I noticed that the blue numbers appear in the degrees of irreducible representations of the Thompson group $Th$, given by the finite sequence of 48 integers (A003916),

$$1, 248, 4123, \color{blue}{27000}, 30628, 30875, 61256, \color{blue}{85995}, 147250, 767637, 779247, 957125, \color{blue}{1707264}, 2450240, 2572752, 3376737, \color{blue}{4096000}, 4123000, 4881384, 4936750,\dots\color{blue}{44330496},\dots 190373976.$$

with repeated terms deleted for brevity.

Questions:

1. If it is not coincidence, what is the reason?
2. If we use other $E_n(\tau)$, would there be "connections" to other sporadic groups?

I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2,

$$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, \eta^2(4\tau)\,\sqrt[3]{j(4\tau)}$$

which has the nice q-expansion (A027652, negated terms),

$$g(\tau) = 1/q - 2 + 248q^3 - 492q^4 +(15^3+744)q^7 + \dots + (5280^3+744)q^{67} + \dots + (640320^3+744)q^{163}+\dots$$

However, one can use other Eisenstein series $E_n(\tau)$ as the one below.

II. In a paper by Bruinier (p.6), Borcherds defines a modular form of weight 1/2. First let,

$$K(\tau) = \tfrac{1}{16}\big(\vartheta_3^4(\tau) - \tfrac{1}{8}\vartheta_2^4(\tau)\big)\,\vartheta_2^4(\tau)\vartheta_3(\tau)\vartheta_4^4(\tau)$$

then,

$$b(\tau) = 60\vartheta_3(\tau)+\frac{K(\tau)E_6(4\tau)}{\eta^{24}(4\tau)} = 60\vartheta_3(\tau)+\frac{K(\tau)\sqrt{j(4\tau)-1728}}{\eta^{12}(4\tau)} $$

This has the q-expansion (A013953),

$$b(\tau) = 1/q^3 + 4 - 240q + (\color{blue}{27000}-240)q^4 - \color{blue}{85995}q^5 + \color{blue}{1707264}q^8 - (\color{blue}{4096000}+240)q^9 + \color{blue}{44330496}q^{12}-91951146q^{13}+\dots$$

I noticed that the blue numbers appear in the degrees of irreducible representations of the Thompson group $Th$, given by the finite sequence of 48 integers (A003916),

$$1, 248, 4123, \color{blue}{27000}, 30628, 30875, 61256, \color{blue}{85995}, 147250, 767637, 779247, 957125, \color{blue}{1707264}, 2450240, 2572752, 3376737, \color{blue}{4096000}, 4123000, 4881384, 4936750,\dots\color{blue}{44330496},\dots 91171899, 111321000, 190373976.$$

with repeated terms deleted for brevity.

Questions:

1. If it is not coincidence, what is the reason?
2. If we use other $E_n(\tau)$, would there be "connections" to other sporadic groups?