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A030205 Expansion of q^(-1/2) * eta(q)^2 * eta(q^5)^2 in power of q.

%I A030205 #63 Aug 06 2025 17:18:48
%S A030205 1,-2,-1,2,1,0,2,2,-6,-4,-4,6,1,4,6,-4,0,-2,2,-4,6,-10,-1,-6,-3,12,-6,
%T A030205 0,8,12,2,2,-2,2,-12,-12,2,-2,0,8,-11,6,6,-12,-6,4,8,4,2,0,6,14,4,-6,
%U A030205 2,-4,-6,-6,2,-12,-11,-12,-1,2,20,0,-8,-4,18,-4,12,0
%N A030205 Expansion of q^(-1/2) * eta(q)^2 * eta(q^5)^2 in power of q.
%C A030205 This eta-quotient of conductor 20 is one of the twelve weight 2 newforms listed by Martin and Ono.
%C A030205 The associated elliptic curve is "20a1": y^2 = x^3 + x^2 + 4*x + 4 or "20a2": y^2 = x^3 + x^2 - x.
%C A030205 Number 39 of the 74 eta-quotients listed in Table I of Martin (1996).
%C A030205 The mentioned eta-quotient is in fact eta^2(2*z) * eta^2(10*z) with q = exp(2*Pi*i*tau) with Im(tau) > 0, I^2 = -1, with the q-expansion coefficients b(n) from the Michael Somos Oct Aug 13 2006 formula: b(2*n) = 0 and b(2*n+1) = a(n), for n >= 0. A273163(k) = b(prime(k)), k >= 1. See also the comments on multiplicativity of b(n) (called there c(n)) with b(2^k) = b(2)^k = 0, b(5^k) = b(5)^k = (-1)^k, and b(prime(n)^k) = (sqrt(prime(n)))^k*S(k,A273163(n)/sqrt(prime(n))) with Chebyshev's S polynomials (A049310), for n = 2, and n >= 4 and k >= 2. Compare this with the b(p^(e+2)) recurrence given by Michael Somos, Oct 31 2005. - _Wolfdieter Lang_, May 23 2016
%H A030205 Seiichi Manyama, <a href="/A030205/b030205.txt">Table of n, a(n) for n = 0..1000</a>
%H A030205 M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787564">On McKay's conjecture</a>, Nagoya Math. J., 95 (1984), 85-89.
%H A030205 LMFDB, <a href="https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/20/2/">Space of modular forms of level 20 and weight 2</a>.
%H A030205 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 A030205 Y. Martin and K. 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 (1997), no. 11, 3169-3176. MR1401749 (97m:11057)
%H A030205 K. Ono, <a href="http://www.jstor.org/stable/2974471">Ramanujan, taxicabs, birthdates, ZIP codes and twists</a>, Amer. Math. Monthly, 104 (1997), 912-917. MR1490909 (98i:11020)
%H A030205 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%F A030205 Euler transform of period 5 sequence [ -2, -2, -2, -2, -4, ...]. - _Michael Somos_, Oct 31 2005
%F A030205 Given g.f. A(x), then B(x) = q * A(q)^2 satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u*w * (u + 8*v + 16*w) - v^3. - _Michael Somos_, Oct 31 2005
%F A030205 a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(5^e) = (-1)^e, else b(p^(e+2)) = b(p)*b(p^(e+1)) - p*b(p^e). - _Michael Somos_, Oct 31 2005
%F A030205 a(n) = b(2*n + 1) and b(p) = p minus number of points of elliptic curve "20a1" or "20a2" modulo p. - _Michael Somos_, Aug 13 2006
%F A030205 G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(5*k)))^2.
%F A030205 a(121*n + 60) = -11 * a(n).
%F A030205 Convolution square is A030210. - _Michael Somos_, Jun 13 2014
%F A030205 a(n) = (-1)^n * A159817(n). - _Michael Somos_, Jun 10 2015
%F A030205 G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 20 (t/i)^2 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Jun 10 2015
%e A030205 G.f. = 1 - 2*x - x^2 + 2*x^3 + x^4 + 2*x^6 + 2*x^7 - 6*x^8 - 4*x^9 - 4*x^10 + ...
%e A030205 G.f. of b(n) from eta^2(2*z)*eta^2(10*z) = q - 2*q^3 - q^5 + 2*q^7 + q^9 + 2*q^13 + 2*q^15 - 6*q^17 - 4*q^19 + ..., where q = exp(2*Pi*I*z).
%t A030205 a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^5])^2, {x, 0, n}]; (* _Michael Somos_, May 28 2013 *)
%o A030205 (PARI) {a(n) = if( n<0, 0, n = 2*n + 1; ellan( ellinit( [0, 1, 0, 4, 4], 1), n)[n])}; /* _Michael Somos_, Oct 31 2005 */
%o A030205 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^5 + A))^2, n))}; /* _Michael Somos_, Oct 31 2005 */
%o A030205 (PARI) {a(n) = if( n<0, 0, n = 2*n + 1; ellan( ellinit( [0, 1, 0, -1, 0], 1), n)[n])}; /* _Michael Somos_, Aug 13 2006 */
%o A030205 (PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p==5, (-1)^e, a0=1; a1 = y = -sum( x=0, p-1, kronecker( x^3 + x^2 - x, p)); for( i=2, e, x = y*a1 - p*a0; a0=a1; a1=x); a1)))}; /* _Michael Somos_, Aug 13 2006 */
%o A030205 (Sage) CuspForms( Gamma0(20), 2, prec=92).0; # _Michael Somos_, May 28 2013
%o A030205 (Magma) Basis( CuspForms( Gamma0(20), 2), 145) [1]; /* _Michael Somos_, May 27 2014 */
%o A030205 (Magma) A := Basis( CuspForms( Gamma1(20), 2), 145); A[1] - 2*A[3]; /* _Michael Somos_, May 17 2015 */
%Y A030205 Cf. A030210, A049310, A159817, A273163.
%K A030205 sign
%O A030205 0,2
%A A030205 _N. J. A. Sloane_