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Remember the Fourier expansion of the full modular form: $$ \Delta(z) = \sum_{n=1} \tau(n)q^n $$ where $\tau(n)$ are the coefficients (Ramanujan Tau function). Ramanujan found that the coefficients satisfy $$ \tau(n) \equiv \sigma_{11}(n) \mod 691 $$ where $\sigma_{11}(n) = \sum_{d|n}d^{11}$. My question:

Do you know other modular forms which Fourier expansion coefficients satisfy such a congruence?

With such I mean this: Let $f(z)$ be a modular form so that $f(z)=\sum a_nq^n$. Now I want to find a congruence such as $$ a_n \equiv g(n) \mod m $$ where $g(n)$ is a (simple) arithmetic function such as the divisor sum function $\sigma(n)$ and $m$ favorable a prime.

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  • $\begingroup$ These things are pretty common. What's really going on here is that $\Delta$ is congruent to the Eisenstein series $E_{12}$ modulo $691$. If $f$ is any modular eigenform, and $\rho_l:\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)\to\mathrm{GL}_2(\overline{\mathbb Q}_l)$ is its associated $l$-adic Galois representation, then a congruence of this type will exist whenever $\rho_l$ is reducible modulo $l$. Whilst for any given $f$, this can only happen for finitely many primes, it's pretty common that it will happen for at least one. $\endgroup$
    – Mathmo123
    Sep 26, 2017 at 16:25
  • $\begingroup$ Thanks, can you give me an example? $\endgroup$
    – Fabian Schn.
    Sep 26, 2017 at 16:59
  • $\begingroup$ Kilford's book has a chapter about modular forms $\bmod p$. $\endgroup$
    – reuns
    Sep 26, 2017 at 17:21

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One example is $f(z)=\eta(3\tau)^9/\eta(\tau)^3= q + 3q^2 + 9q^3 + 13q^4 + 24q^5 + 27q^6 + \dots\;$ where $a_n=g(n)=\sum_{d|n}d^2\;\big($ $\!\!n/d\over{3}$ $\!\!\big)\;$ and $\;(\;.\mid 3)$ is a Kronecker symbol. Now $a_n \equiv g(n) \pmod m$ for any integer $m$. See OEIS sequence A106402 for more details of this modular form.

Other examples are in this Mathoverflow question about sequences A001935 and A001936. $\sigma(8n+1) \equiv A001935(n) \pmod 4$ and $\sigma(4n+1) \equiv A001936(n) \pmod 4.$

Related example $f(z)=\eta(4\tau)^8/\eta(2\tau)^4=q +4q^3 +6q^5 +8q^7 +13q^9 +12q^{11}+\dots\;$ where $a_n=\sigma(n)\;$ for all odd $n$. Another exmaple is this Mathoverflow question about sequence A145722 where $\;\sigma(4n-3) \equiv A145722(n-1) \pmod 5.$

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  • $\begingroup$ And what is this congruent to? $\endgroup$
    – Fabian Schn.
    Sep 26, 2017 at 20:14
  • $\begingroup$ How is $F(s) = \zeta(s-2) L(s,\chi_3)$ is the L-series of a modular form ? Ah I got it : it is because $F(3-s)$ is (up to some gamma factors) again a Dirichlet series. $\endgroup$
    – reuns
    Sep 26, 2017 at 21:17

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