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

a citation would be useful to me,

I want to check my calculation and

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.