Reference request: theta functions for lattices which are not unimodular or even

Question

In A Course in Arithmetic, Serre works out some of the theory of theta functions for even, self-dual lattices, e.g. such theta functions are modular forms of weight $n/2$, where $n$ is the dimension of the lattice (necessarily $8|n$).

I want to learn about theta functions for more general lattices, which are modular forms for congruence subgroups of $SL_2(\mathbb{Z})$. Specifically, theorems such as "For lattices satisfying __ the corresponding theta function is a modular form for the subgroup __" would be fantastic. Can anybody suggest references for this material?

Thanks in advance.

Answer 1

Some coverage of this is given in Chapter 1 of Peter Sarnak's Some Applications of Modular Forms.