## Non-ordinary primes: A story.(English)Zbl 0887.11020

This paper deals with normalized eigenforms of level 1 with Fourier expansions of the form $f=q+ \sum^\infty_{n=2} a_nq^n$ with integer coefficients $$a_n$$, where $$q= e^{2\pi iz}$$ and $$\text{Im} (z)>0$$. If a prime $$p$$ divides the $$p$$th coefficient $$a_p$$, the form $$f$$ is called ordinary at $$p$$. (The author also refers to $$p$$ as an ordinary prime of $$f$$.) In earlier work, D. Niebur [Ill. J. Math. 19, 448-449 (1975; Zbl 0301.10025)] treated the classic discriminant $$\Delta= \sum^\infty_{n=1} \tau(n) q^n$$, where $$\tau(n)$$ is Ramanujan’s tau function, by using an arithmetical identity expressing $$\tau(n)$$ in terms of $$\sigma (n)$$, the sum of the divisors of $$n$$. Niebur found that the only non-ordinary primes for $$\Delta$$ between 2 and 65064 are 2, 3, 5, 7 and 2411.
The present author considers the unique normalized eigenforms $$\Delta_k$$ of weight $$k$$ where $$k\in \{12, 16, 18, 20, 22, 24\}$$, $$(\Delta_{12} = \Delta)$$ and obtains generalizations of Niebur’s formula that are amenable to computation, and then uses these to determine all the non-ordinary primes $$p\leq 10^6$$ for each $$\Delta_k$$.

### MSC:

 11F03 Modular and automorphic functions 11F33 Congruences for modular and $$p$$-adic modular forms

Zbl 0301.10025
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### References:

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