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A187076 Coefficients of L-series for elliptic curve "144a1": y^2 = x^3 - 1.

%I A187076 #42 Feb 16 2025 08:33:14
%S A187076 1,4,2,-8,-5,4,-10,-8,9,0,14,16,-10,4,0,8,14,-20,2,0,-11,-20,-32,16,0,
%T A187076 4,14,-8,-9,-20,26,0,2,28,0,16,16,28,-22,0,14,-16,0,-40,0,28,26,-32,
%U A187076 -17,0,-32,16,-22,0,-10,-32,-34,8,14,0,45,4,38,-8,0,0,-34
%N A187076 Coefficients of L-series for elliptic curve "144a1": y^2 = x^3 - 1.
%C A187076 Number 67 of the 74 eta-quotients listed in Table I of Martin (1996).
%C A187076 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A187076 Andrew Howroyd, <a href="/A187076/b187076.txt">Table of n, a(n) for n = 0..1000</a>
%H A187076 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 A187076 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 A187076 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H A187076 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A187076 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A187076 Expansion of f(x)^4 in powers of x where f() is a Ramanujan theta function.
%F A187076 Expansion of q^(-1/6) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^4 in powers of q.
%F A187076 Euler transform of period 4 sequence [4, -8, 4, -4, ...].
%F A187076 G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = f(t) where q = exp(2 Pi i t).
%F A187076 a(n) = b(6*n + 1) and b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)), b(p) = 0 if p == 5 (mod 6), b(p) = 2 * x where p = x^2 + 3 * y^2 == 1 (mod 6) and x == 4, 5 (mod 6).
%F A187076 G.f.: Product_{k>0} (1 - (-x)^k)^4. a(n) = (-1)^n * A000727(n).
%F A187076 Convolution cube is A209676. - _Michael Somos_, Jun 10 2015
%F A187076 a(2*n) = A258779(n). a(2*n + 1) = 4 * A187150(n). - _Michael Somos_, Jun 10 2015
%e A187076 G.f. = 1 + 4*x + 2*x^2 - 8*x^3 - 5*x^4 + 4*x^5 - 10*x^6 - 8*x^7 + 9*x^8 + ...
%e A187076 G.f. = q + 4*q^7 + 2*q^13 - 8*q^19 - 5*q^25 + 4*q^31 - 10*q^37 - 8*q^43 + 9*q^49 + ...
%t A187076 a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^4, {x, 0, n}]; (* _Michael Somos_, Jun 10 2015 *)
%o A187076 (PARI) {a(n) = if( n<0, 0, ellak( ellinit( [0, 0, 0, 0, -1], 1), 6*n + 1))};
%o A187076 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A) * eta(x^4 + A)))^4, n))};
%o A187076 (PARI) {a(n) = my(m, A, p, e, x, y, a0, a1); if( n<0, 0, m = 6*n + 1; A = factor(m); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, 0, p%6==5, if(e%2, 0, (-p)^(e/2)), for( y=1, sqrtint(p\3), if( issquare(p - 3*y^2, &x), break)); a0 = 1; if( x%6>3, x = -x); a1 = x = 2*x; for( i=2, e, y = x*a1 - p*a0; a0 = a1; a1 = y); a1)))};
%o A187076 (Magma) A := Basis( CuspForms( Gamma0(144), 2), 398); A[1] + 4*A[7] + 2*A[11] - 8*A[13]; /* _Michael Somos_, Jan 01 2017 */
%Y A187076 Cf. A000727, A187150, A209676, A258779, A280384.
%K A187076 sign
%O A187076 0,2
%A A187076 _Michael Somos_, Mar 05 2011