Hello, I was trying to solve Exercise 1.4.5 in Alvaro Lozano-Robledo's book Elliptic Curves, Modular Forms and Their L-functions, which is about representations of integers as sums of 6 squares and its relation to the theta function
$$\Theta(q) = \sum_{j = -\infty}^{\infty} q^{j^2} $$
I need to define the space of modular forms $M_3(\Gamma_1(4))$ in SAGE, which I already did and find a basis for this 2-dimensional space. I was able to this without any problems.
But now I'm asked to write $\Theta^6(q)$ as a linear combination of the basis elements just found. This prompts me to ask some questions.
1) How do I define $\Theta(q)$ and how do I check that $\Theta^6(q) \in M_3(\Gamma_1(4))$?
2) How would I express $\Theta^6(q)$ as a linear combination of the basis elements?
3) More generally, is there a way in which one can specify some q-series expansion and ask SAGE if it is in a particular space of modular forms and if it is to express it as a linear combination of the basis elements?
I've already searched in the SAGE manual but I only found how to define Eisenstein series and the like. I apologize if my questions are not very well formulated.
Thank you very much in advance for any help.
