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I am looking for a reference for the transformation formulae for the classical theta-functions $$\theta_4(\tau)=\sum_{n=-\infty}^\infty (-1)^n q^{n^2}$$ and $$\theta_2(\tau)=\sum_{n=-\infty}^\infty q^{(2n+1)^2/4}$$ under the congruence group $\Gamma_0(4)$. Here $\tau$ lies in the upper-half plane and $q^x$ denotes $\exp(2\pi i x\tau)$. More precisely I want the exact automorphy factors for each $A\in\Gamma_0(4)$ (some eighth root of unity times $\sqrt{c\tau+d}$). I know these can easily be deduced from those for the basic theta-function $$\theta_3(\tau)=\sum_{n=-\infty}^\infty q^{n^2}$$ for which a nice reference for the automorphy factors is Koblitz's Introduction to Elliptic Curves and Modular Forms. However

1. a citation would be useful to me,

2. I want to check my calculation and

3. a reference may give the formulae in a more convenient form than I have.

Thanks in advance.

EDIT I have now found a convenient reference: Rademacher's Topics in Analytic Number Theory.

FURTHER EDIT Rademacher atcually gives full transformation formula for the two-variable classical Jacobi theta functions under arbitrary matrices in $\mathrm{SL}_2(\mathbb{Z})$. From these we can deduce for $A\in\Gamma_1(4)$ that $$\frac{\theta_2(A\tau)}{\theta_3(A\tau)} =i^b\frac{\theta_2(\tau)}{\theta_3(\tau)}$$ and $$\frac{\theta_4(A\tau)}{\theta_3(A\tau)} =i^{-c/4}\frac{\theta_4(\tau)}{\theta_3(\tau)}$$ in the usual notation. Once noticed, these relations are easy to prove from scratch.

Thanks to all who replied to this question.

3 Included answer

I am looking for a reference for the transformation formulae for the classical theta-functions $$\theta_4(\tau)=\sum_{n=-\infty}^\infty (-1)^n q^{n^2}$$ and $$\theta_2(\tau)=\sum_{n=-\infty}^\infty q^{(2n+1)^2/4}$$ under the congruence group $\Gamma_0(4)$. Here $\tau$ lies in the upper-half plane and $q^x$ denotes $\exp(2\pi i x\tau)$. More precisely I want the exact automorphy factors for each $A\in\Gamma_0(4)$ (some eighth root of unity times $\sqrt{c\tau+d}$). I know these can easily be deduced from those for the basic theta-function $$\theta_3(\tau)=\sum_{n=-\infty}^\infty q^{n^2}$$ for which a nice reference for the automorphy factors is Koblitz's Introduction to Elliptic Curves and Modular Forms. However

1. a citation would be useful to me,

2. I want to check my calculation and

3. a reference may give the formulae in a more convenient form than I have.

Thanks in advance.

EDIT I have now found a convenient reference: Rademacher's Topics in Analytic Number Theory.

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