Apologies for the vague title but I wasn't sure how to word this. To preface my question, let's recall the theta function:
$$\theta(\tau) = \sum_{n \in \mathbb Z} e^{i \pi n^2 \tau}$$
This function is interesting because the $n$th coefficient of $\theta^k$ as a series of terms of the form $e^{i \pi n ^2 \tau}$ is the number of ways to write $n$ as a sum of $k$ squares. From the course I am currently taking, it turns out that if $k$ is a multiple of $8$, then $\theta \in M_{k/2}(\Gamma)$ where $\Gamma$ is a particular congruence subgroup of $SL_2(\mathbb Z)$. From this, some nice information about the coefficients can be deduced due to known theory about modular forms.
Now suppose we consider the family of functions
$$\eta_r = \sum_{p \text{ prime}} e^{i \pi p^r \tau}$$
These functions would be of interest as, similar to above, the $n$th coefficient of $\eta_r^k$ written as a power series is the number of ways to write $n$ as a sum of $k$th powers of primes, which is relevant to the Waring-Goldbach problem.
My question is, are any of the $\eta_r$ modular forms? Realistically, I am looking for a proof that they aren't, because it's rather far-fetched that they would be. So far all I have been able to prove is that if they were of level $\Gamma \supset \Gamma(N)$ then $N$ would have to be even. Any advance on this would be nice. Thanks.