I have a modular form I am constructing out of sums and products of various dissected divisor-sum series, namely forms of the type $$f_i = \sum_{j=0}^\infty \sigma_1(36j+i) q^{36j+i}.$$

Each of these can be constructed from the standard forms $\sum \sigma_1(2n+1) q^{2n+1}$ and $\sum \sigma_1(3n+1) q^{3n+1}$ by various quadratic twists and magnifications (applications of $U(d)$ and $V(d)$). (Note: this increases the level by factors up to $36^2$, so if there is a better way to do that I'm all ears.)

I'd like to use Sturm's theorem on the resulting form, but if I understand correctly (I certainly might not!), summing modular forms in $M_k(\Gamma_0(N_1),\chi)$ and $M_k(\Gamma_0(N_2),\psi)$ means that I am verifiably in $M_k(\Gamma_1(lcm(N_1,N_2)),\phi)$ for some $\phi$, and not $M_k(\Gamma_0(lcm(N_1,N_2)),\phi)$. The Sturm bound for $\Gamma_1$ is substantially larger than for $\Gamma_0$, possibly infeasibly so. Arguments I have seen involving similar sums in the prior literature don't seem to worry about that, though.

Is there an argument for this restricted case that I am still in $\Gamma_0$, or am I forced to move to $\Gamma_1$?

Lacking that, is there a better bound I can state on, say, $\sum \sigma_1(36j+2) q^{36j+2}$ than $N=5832 = 9 \cdot 18^2 \cdot 2$? ($3n+1$ has original weight 9, quadratic twist mod 18 to get $18n+1$, magnify by 2 to get $36n+2$?)

I have Ono's "Web" personally and Koblitz in the library, and can try for any other citations via interlibrary loan. Many thanks.