The theta function of an odd Dirichlet character

Question

The theta function $\theta_\chi(t)$ of a Dirichlet character $\chi$ is defined to be $\theta_\chi(t) = \frac{1}{2} \sum\limits_{n=-\infty}^\infty \chi(n) e^{2\pi i n^2 t}$ if $\chi(-1) = 1$ (i.e., $\chi$ is even), or $\frac{1}{2} \sum\limits_{n=-\infty}^\infty n\chi(n) e^{2\pi i n^2 t}$ if $\chi(-1) = -1$ (i.e., $\chi$ is odd).

I understand the definition when $\chi$ is odd can't be the same as the definition when $\chi$ is even because the formula would have vanished identically. Multiplying an additional factor of $n$ indeed could solve the problem since $n\chi(n)$ is once again even.

Is there a more intrinsic reason why the theta function for an odd Dirichlet character is twisted in this specific way? Theoretically speaking, one can twist by any odd function of $n$. Is there a deeper reason why this form of twist is preferred over others?

Answer 1

The point is that $\theta_\chi(t)$ is a modular form whose coefficients are essentially $\chi(n)$. If we twist $\chi(n)$ by some random odd smooth function of $n$, we don't get a modular form. (Perhaps there are some examples other than twisting by $n$, but they will be artificial. That is, in a sense, $\theta_\chi(t)$ is the only natural example obtainable this way.)