I did some calculations on coefficients of $q$-expansion of element of $SL_2(\Gamma_0(19))$ with $a_1=1$, and I think the coefficients satisfies $a_{mn}=a_ma_n$ if $\gcd(m,n)=1$. Does there exist any literature that contains the proof of this (or more general) fact?
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Maybe you can find interesting things here ? In particular pages 242 and 256. – Jean Marie Jan 11 '23 at 11:00
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@JeanMarie thanks for the reference. I will take a closer look to that – Laurence PW Jan 11 '23 at 11:20
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2What do you call $SL_2(\Gamma_0(19))$? But I think any course or text on modular forms will prove this (in fact, a wide generalization). Look out for the keyword Hecke operators and eigenforms. – Aphelli Jan 11 '23 at 11:23
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@Dietrich Burde: $\mathcal{S}_2(\Gamma_0(19))$ (which I think the OP is interested about) has dimension one. – Aphelli Jan 11 '23 at 12:18
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@Aphelli actually it is one of the exercise that I got, I didn't it is related to Hecke operators – Laurence PW Jan 11 '23 at 13:18
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2Hecke operators were basically introduced to explain why this kind of multiplicative property happened. – Aphelli Jan 11 '23 at 13:20
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@Aphelli ok thank you, I will take a look to some literature. – Laurence PW Jan 11 '23 at 14:09
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You may have a look at the proof for Ramanujan Tau function : https://math.stackexchange.com/a/2666087/72031 – Paramanand Singh Jan 12 '23 at 11:55
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It is true that the cuspform of weight $2$ on $\Gamma_0(19)$, normalized with $a(1) = 1$, has multiplicative coefficients.
I write the because the vector space of weight $2$ modular forms on $\Gamma_0(19)$ has dimension $2$, and the cuspidal subspace has dimension $1$. Thus there is exactly one, and its $q$-expansion begins $$ q - 2q^3 - 2q^4 + 3q^5 - q^7 + q^9 + 3q^{11} + 4q^{12} - 4q^{13} + O(q^{15}). $$
The general theory is that for any congruence subgroup of $\mathrm{SL}(2, \mathbb{Z})$ (such as $\Gamma_0(19)$), there are particular linear operators called Hecke operators. These operators take cusp forms to cusp forms, hence we can consider their action on the subspace of cusp forms. These operators also commute, and each Hecke operators commutes with its adjoint operator (with respect to the Petersson inner product).
The spectral theorem from linear algebra then guarantees that there is a basis for the space of cusp forms consisting of simultaneous eigenvectors. Computing the action of these Hecke operators shows that a simultaneous eigenform has multiplicative coefficients.
The result is that there is a basis of cuspidal eigenforms for any fixed weight and congruence subgroup. As the space of cuspforms of weight $2$ on $\Gamma_0(19)$ is one-dimensional, it must be the basis and is thus an eigenform --- and hence has multiplicative coefficients.
This result is fundamental in the theory of modular forms, and any introductory book on modular forms should include it. Diamond and Shurman's introduction is good (this result is one of the major pieces of chapter 5 of that book). Bumps Automorphic Forms and Representations includes a concise proof for level $1$ in one section of the first chapter, though a beginner to to the subject might find it too terse. (At least I thought so when I was first learning, though I like it a lot now).
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