What relations exist among (quasi-) modular forms of different levels?

Question

Given a (quasi-) modular form $f(\tau)$ for some congruence subgroup (say) $\Gamma(k)$, we know that $f(N\tau)$ is a (quasi-) modular form for $\Gamma(N k)$. Is there anything known about when we can do a partial reverse, that is, when we can take linear combinations of (quasi-) modular forms for some higher level subgroup to obtain one of stricktly lower level?

An example is the following:

Let $E(q) = \sum_{k=0}^\infty \sigma_1(2k+1)q^{2k+1}$ and $A(q) = \sum_{k=1}^\infty \sigma_1(k)q^k$. Then $E(q)$ is modular with respect to a non-trivial character, and both $A(q^2)$ and $A(q^4)$ are quasi-modular of level 2 and 4, respectively (though not of pure weight).

However: it turns out that $$E(q) + 3A(q^2) - 2A(q^4) = A(q)$$ which shows that a linear combination of higher level terms (and one which is modular with respect to a non-trivial character) yields one of lower level.

Is this simply random chance? Are there known relations of this type?

Answer 1

There are (at least) two answers: there is a Galois theory of modular functions (as in G. Shimura's 1971 book, or in Lang's book on elliptic functions), and a theory of newforms" (Atkin-Lehner, and also Casselman in a representation-theoretic context).

Thinking in terms of general Galois theory, it ought not be so surprising that various sums of non-invariant things become invariant, I suppose, but the particulars are often non-trivial.