So I am (barely) familiar with the construction of the theta function of an integral lattice $L$. The theta function, as I understand it, is defined as the function which takes a variable $z$ and spits out the sum of $q$ raised to the $N(v)$ power, where the sum is over all vectors $v$ in the lattice, $q$ is equal to $exp(2 \pi i z)$, and $N(v)$ is the norm squared. We can regard this as a formal sum for the sake of this question. Let's call it $T(z)$.


I know there all kinds of identities about this function. For example if the lattice is even and unimodular, then $T(z)$ is a modular form. I am curious about the related function $T(-z)$. Is there an easy way to relate it to $T(z)$? 


What I'm really really interested in is the quotient $T(z)/T(-z)$. I know there are people who know this stuff way better then I do, can anyone help? Is it easier in the unimodular case?



Theta Functions and Cousins
I'm having trouble understanding this, because under your definition, T(-z) requires T to be defined on the lower half-plane. The series doesn't converge there.


Also, N(v) should be half of the squared norm. Equivalently, N(v) should be the value of the quadratic form that defines the inner product on the lattice.


I can think of one interpretation, where the theta function is actually a [Jacobi theta function](http://en.wikipedia.org/wiki/Jacobi_theta_functions#Jacobi_theta_function), which is a [Jacobi form](http://en.wikipedia.org/wiki/Jacobi_form), i.e., a section of a line bundle on the universal elliptic curve. This can be viewed as a function T(t,z) on HxC, with some invariance properties under translation by lattice elements in C and SL(2,Z) transformations in H. Then you can negate the z variable.


Eichler and Zagier have a book on Jacobi forms, called *The theory of Jacobi forms*.


