# How to define an element in a space of Modular Forms and express it as a linear combination of basis elements?

 0 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. asked Jan 28 '12 Adrián Barquero 1 ● 1

 0 It is not really a sage question, so I'll pass (let's just say that it's a theta series, so there's a quadratic form hidden). You just need $\Theta(q)=a+bq+O(q^2)$, because then in terms of basis given by sage, it will be $a$ times the first plus $b$ times the second. More generally, sage gives basis in a form that makes it pretty easy to read a conjectural linear combination from the first terms of the expansion ; if you already know you got the expansion from a modular forms in the right space, that will be enough to know the equality. EDIT: you might be interested by the method find_in_space from the modular spaces objects. posted Jun 21 '12 Snark 1 ● 2

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