1 | initial version |

- 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.

2 | Add something about find_in_space |

- 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.

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