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Hi, I've been thinking about this questions:

Is there a modular form in $M_2 (\Gamma_0(11))$ which only vanishes at the cusp 0? If so how can we find it?

Edit: There was another question which I removed.

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Yes, take a suitable linear combination of the Eisenstein form with the cuspform $\eta(q)^2 \eta(q^{11})^2$. – Noam D. Elkies Jan 12 '12 at 16:36
This cuspform vanishes at 0, does the Eisenstein form vanish at 0 as well? Also, how do I make sure the linear combination does not vanish anywhere else on the upper half plane? – Nadim Rustom Jan 12 '12 at 17:01
Sorry, I didn't notice that you asked for no other zeros at all (though come to think of it what I wrote doesn't make sense anyway for the reason you suggest). OK, let's try again: there is no such form $\phi$, because if it existed then $\phi / (\eta(q)^2 \eta(q^{11}))^2$ would be a function on degree 1 on $X_0(11)$ [pole at the cusp $\infty$, zero at the cusp $0$], which is impossible because $X_0(11)$ is known to have genus 1. – Noam D. Elkies Jan 12 '12 at 17:12
Alternatively, the space of cusp forms has codimension one, and the Atkin-Lehner involution switches the two cusps. Thus, any form must either vanish at both cusps or vanish at neither. – Kevin Ventullo Jan 12 '12 at 20:09
@Nadim : What do you mean exactly by vanishing ? Do you mean that there is no constant term in the Fourier expansion (at the cusp 0), or do you mean vanishing as a differential form on $X_0(11)$ ? This is not the same, for example the cusp form $\eta(q)^2 \eta(q^11)^2 \frac{dq}{q}$ doesn't vanish anywhere as a differential form. – François Brunault Jan 12 '12 at 20:57

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