A271231 Expansion of the modular cusp form ( eta(q^4) * eta(q^12) )^4 / ( eta(q^2) * eta(q^6) * eta(q^8) * eta(q^24) ), where eta is Dedekind's eta function.

0, 1, 0, 1, 0, -2, 0, 0, 0, 1, 0, -4, 0, -2, 0, -2, 0, 2, 0, 4, 0, 0, 0, 8, 0, -1, 0, 1, 0, 6, 0, -8, 0, -4, 0, 0, 0, 6, 0, -2, 0, -6, 0, -4, 0, -2, 0, 0, 0, -7, 0, 2, 0, -2, 0, 8, 0, 4, 0, -4, 0, -2, 0, 0, 0, 4, 0, 4, 0, 8, 0, -8, 0, 10, 0, -1, 0, 0, 0, 8, 0, 1, 0, 4, 0, -4, 0, 6, 0, -6, 0, 0, 0, -8, 0, -8, 0, 2, 0, -4, 0, -18, 0, -16

Offset

0,6

Comments

The modularity pattern of the elliptic curve y^2 = x^3 + x^2 + x considered modulo prime(m) is seen from a(prime(m)) = prime(m) - N(prime(m)) = A271230(m), where N(prime(m))= A271229(m) is the number of solutions of this congruence. That is, the p-defect coincides with the prime indexed expansion coefficient (here for all primes).

This modular cusp form of weight 2 and level N = 48 = 2^4*3 is Nr. 54 in Martin's Table 1 (corrected by giving the 24 the missing exponent -1). See also the Michael Somos link where this correction has been observed.

This modular cusp form is a simultaneous eigenform of every Hecke operators T_p, with p a prime not 2 or 3 (bad primes) with eigenvalue lambda(p) = a(p). (See the Martin reference, Proposition 33, p. 4851.)

In the Martin and Ono reference, p. 3173 (Theorem 2), this cusp form appears (in the corrected version) in the row Conductor 48, and it is there related to the elliptic curve y^2 = x^3 + x^2 - 4*x - 4. The p-defects of this curve coincide with the ones of the curve y^2 = x^3 + x^2 + x modulo primes p given in A271230. - Wolfdieter Lang, Apr 21 2016

Multiplicative. See A159819 for formula. - Andrew Howroyd, Aug 06 2018

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Formula

a(2*n+1) = A159819(n), a(2*n) = 0.

O.g.f.: Expansion in q = exp(2*Pi*i*z) with Im(z) > 0 of (eta(4*z)*eta(12*z))^4 / (eta(2*z)*eta(6*z)*eta(8*z)*eta(24*z)), where eta(z) = q^(1/24)*Product_{n >= 1} (1 - q^n) is the Dedekind function with q = q(z) given above, and i is the imaginary unit.

a(prime(m)) = A271230(m), m >= 1.

Example

n=2: a(2) = A271230(1) = 0.
n=5: a(5) = A271230(3) = -2.
See the example section of A271229 for the solutions for the first primes.

Mathematica

QP = QPochhammer;
a[n_] := If[OddQ[n], SeriesCoefficient[QP[-x] QP[x^2] QP[-x^3] QP[x^6], {x, 0, (n-1)/2}], 0];
a /@ Range[0, 100] (* Jean-François Alcover, Sep 19 2019 *)

Prog

(PARI) q='q+O('q^220); concat([0], Vec( (eta(q^4)*eta(q^12))^4 / (eta(q^2)*eta(q^6)*eta(q^8)*eta(q^24) ) ) ) \\ Joerg Arndt, Sep 12 2016

Crossrefs

Cf. A159819, A271229, A271230.

Sequence in context: A227761 A037188 A276847 * A306798 A086079 A296338

Adjacent sequences: A271228 A271229 A271230 * A271232 A271233 A271234

Keyword

sign,easy,mult

Author

Wolfdieter Lang, Apr 19 2016

Status

approved