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A138515 Expansion of q^(-1/4) * eta(q^2)^8 / (eta(q) * eta(q^4))^2 in powers of q.

%I A138515 #43 Feb 16 2025 08:33:08
%S A138515 1,2,-3,-6,2,0,-1,10,0,2,10,-6,-7,-14,0,10,-12,0,-6,0,9,4,10,0,18,2,0,
%T A138515 -6,-14,18,-11,-12,0,0,-22,0,20,-14,-6,-22,0,0,23,26,0,18,4,0,-14,2,0,
%U A138515 20,0,0,0,-12,3,-30,26,0,-30,-14,0,0,2,-30,-28,26,0,18,10,0,-13,34,0,0,20,0,26,-22,0,6,0,-6,18,0
%N A138515 Expansion of q^(-1/4) * eta(q^2)^8 / (eta(q) * eta(q^4))^2 in powers of q.
%C A138515 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A138515 Number 58 of the 74 eta-quotients listed in Table I of Martin (1996). - _Michael Somos_, Mar 16 2012
%C A138515 The weight 2 eta-quotient newform  eta^8(8*z) / (eta^2(4*z)*eta^2(16*z)) appears in Theorem 2 of the Martin and Ono link in the row with conductor 64 for the strong Weil curve y^2 = x^3 - 4*x. For N(p), the number of solutions modulo primes for this elliptic curve and for y^2 = x^3 + x, see A095978. The non-vanishing p-defects p - N(p) for these two curves are given in A267859.  - _Wolfdieter Lang_, May 26 2016
%H A138515 Seiichi Manyama, <a href="/A138515/b138515.txt">Table of n, a(n) for n = 0..10000</a>
%H A138515 Amanda Clemm, <a href="http://www.mdpi.com/2227-7390/4/1/5">Modular Forms and Weierstrass Mock Modular Forms</a>, Mathematics, volume 4, issue 1, (2016)
%H A138515 LMFDB, <a href="http://www.lmfdb.org/EllipticCurve/Q/64/a/4">Elliptic Curve 64.a4 (Cremona label 64a4)</a>
%H A138515 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 A138515 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 A138515 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H A138515 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A138515 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A138515 Coefficients of L-series for elliptic curve "64a4": y^2 = x^3 + x.
%F A138515 Expansion of f(q)^2 * f(-q^2)^2 = psi(-q)^2 * phi(q)^2 = chi(q)^2 * f(-q^2)^4 = psi(q)^2 * phi(-q^2)^2 = f(q)^4 / chi(q)^2 = f(q)^6 / phi(q)^2 = f(-q^2)^6 / psi(-q)^2 = phi(q)^4 / chi(q)^6 = chi(q)^6 * psi(-q)^4 = f(q)^3 * psi(-q) = f(-q^2)^3 * phi(q) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
%F A138515 Euler transform of period 4 sequence [2, -6, 2, -4, ...].
%F A138515 G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 64 (t/i)^2 f(t) where q = exp(2 Pi i t).
%F A138515 a(n) = b(4*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) if p == 1 (mod 4) with b(p) = 2 * x * (-1)^((x-1)/2) where p = x^2 + 4 * y^2.
%F A138515 G.f.: (Product_{k>0} (1 - x^(2*k))^2 * (1 + x^(2*k - 1)))^2.
%F A138515 a(n) = (-1)^n * A002171(n). a(9*n + 2) = -3 * a(n), a(9*n + 5) = a(9*n + 8) = 0. Convolution square of A138514.
%F A138515 G.f. for{b(n)}:
%F A138515   eta^8(8*z)/(eta^2(4*z)*eta^2(16*z)) with q = exp(2*Pi*i*z)), Im(z) > 0 (see a comment on the Martin-Ono link above). - _Wolfdieter Lang_, May 27 2016
%e A138515 G.f. = 1 + 2*x - 3*x^2 - 6*x^3 + 2*x^4 - x^6 + 10*x^7 + 2*x^9 + 10*x^10 - 6*x^11 + ...
%e A138515 G.f. for {b(n)} = q + 2*q^5 - 3*q^9 - 6*q^13 + 2*q^17 - q^25 + 10*q^29 + 2*q^37 + 10*q^41 - 6*q^45 - 7*q^49 + ...
%t A138515 a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] QPochhammer[ -q])^2, {q, 0, n}]; (* _Michael Somos_, May 15 2015 *)
%t A138515 a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^4 / (QPochhammer[ q] QPochhammer[ q^4]))^2, {q, 0, n}]; (* _Michael Somos_, May 15 2015 *)
%o A138515 (PARI) {a(n) = ellak( ellinit( [ 0, 0, 0, 1, 0], 1), 4*n + 1)};
%o A138515 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^4 / (eta(x + A) * eta(x^4 + A)))^2, n))};
%o A138515 (PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 4*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p%4==1, forstep( x=1, sqrtint(p), 2, if( issquare( p - x^2), y=x; break)); y = 2 * y * (2 - (y%4)); a0 = 1; a1 = y; for(i=2, e, x = y * a1 - p * a0; a0 = a1; a1 = x); a1, if( e%2==0, (-p)^(e / 2)))))};
%o A138515 (Magma) A := Basis( CuspForms( Gamma0(64), 2), 342); A[1] + 2*A[3]; /* _Michael Somos_, May 15 2015 */
%Y A138515 Cf. A002171, A095978, A138514, A267859, A280339.
%K A138515 sign
%O A138515 0,2
%A A138515 _Michael Somos_, Mar 22 2008