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 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?

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?

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 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?

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 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?