nt.number theory - Transformation formulae for classical theta functions

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
$exp(2pi i xtau)$. More precisely I want the exact automorphy
factors for each $AinGamma_0(4)$ (some eighth root of
unity times $sqrt{ctau+d}$). 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 $AinGamma_1(4)$ that
$$frac{theta_2(Atau)}{theta_3(Atau)} =i^bfrac{theta_2(tau)}{theta_3(tau)}$$
and
$$frac{theta_4(Atau)}{theta_3(Atau)} =i^{-c/4}frac{theta_4(tau)}{theta_3(tau)}$$
in the usual notation. Once noticed, these relations are easy to prove
from scratch.

Thanks to all who replied to this question.

• Next physiology - How does the microbial environment in your gut initiate?
• Previous gr.group theory - Can the symmetric groups on sets of different cardinalities be isomorphic?