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%I #51 Dec 12 2023 18:48:49
%S 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,
%T -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,
%U -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
%N 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.
%C 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).
%C 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.
%C 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.)
%C 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
%C Multiplicative. See A159819 for formula. - _Andrew Howroyd_, Aug 06 2018
%H Seiichi Manyama, <a href="/A271231/b271231.txt">Table of n, a(n) for n = 0..10000</a>
%H Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H Yves Martin and Ken Ono, <a href="http://dx.doi.org/10.1090/S0002-9939-97-03928-2">Eta-Quotients and Elliptic Curves</a>, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.
%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%F a(2*n+1) = A159819(n), a(2*n) = 0.
%F 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.
%F a(prime(m)) = A271230(m), m >= 1.
%e n=2: a(2) = A271230(1) = 0.
%e n=5: a(5) = A271230(3) = -2.
%e See the example section of A271229 for the solutions for the first primes.
%t QP = QPochhammer;
%t a[n_] := If[OddQ[n], SeriesCoefficient[QP[-x] QP[x^2] QP[-x^3] QP[x^6], {x, 0, (n-1)/2}], 0];
%t a /@ Range[0, 100] (* _Jean-François Alcover_, Sep 19 2019 *)
%o (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
%Y Cf. A159819, A271229, A271230.
%K sign,easy,mult
%O 0,6
%A _Wolfdieter Lang_, Apr 19 2016
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