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While working with Fourier expansions of cusp forms of congruence subgroups of the modular group, I observed the following patterns in their prime coefficients.

Let $a(n)$ be the Fourier coefficients of the weight-4 cusp form of $\Gamma_{0}(2)$, and let $b(n)$ be the Fourier coefficients of the weight-2 cusp form of $\Gamma_{0}(4)$. (The OEIS data for these sequences is available here and here respectively.) A small numerical experiment that checked the first 1000 or so primes seems to show that for primes $p>2$:

$$a(p) = p^{3}+1$$

$$b(p) = p+1$$

My questions are:

  1. Are these formulas true, and if so, how does one prove them?
  2. Ramanujan's conjecture states that prime coefficients $\tau(p)$ of the weight-12 cusp form of the modular group satisfy the bound $\left|\tau(p)\right| \leq 2 p^{11/2}$. Since these all pertain to prime coefficients of cusp forms, are these kinds of identities related?

(I apologise if this question is too remedial for this forum. I'm trained as a physicist, not a mathematician, so these kinds of statements are unfamiliar to me. A good reference for learning more about this would be much appreciated, even if a detailed answer is too much to ask for.)

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    $\begingroup$ There are no weight 4/2 cusp forms for those groups. $\endgroup$
    – Kimball
    Commented Oct 1 at 23:56
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    $\begingroup$ To add to @Kimball's comment, these are clearly coefficients of Eisenstein series. $\endgroup$
    – Aurel
    Commented Oct 2 at 8:35
  • $\begingroup$ For A002408, as mentioned in the previous comments, it is not accurate to refer to the source of that sequence as an "8-dimensional cusp form." For one thing is is not a cusp form. For another, the term "dimension" is archaic. These days the term "weight" is standard (and weight is half of what was called dimension). $\endgroup$
    – David Farmer
    Commented Oct 16 at 17:24

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