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A159819
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Coefficients of L-series for elliptic curve "48a4": y^2 = x^3 + x^2 + x.
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5
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1, 1, -2, 0, 1, -4, -2, -2, 2, 4, 0, 8, -1, 1, 6, -8, -4, 0, 6, -2, -6, -4, -2, 0, -7, 2, -2, 8, 4, -4, -2, 0, 4, 4, 8, -8, 10, -1, 0, 8, 1, 4, -4, 6, -6, 0, -8, -8, 2, -4, -18, -16, 0, 12, -2, 6, 18, -16, -2, 0, 5, -6, 12, 8, -4, 4, 0, -2, -6, 12, 0, 8, -12
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OFFSET
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0,3
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COMMENTS
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Number 54 of the 74 eta-quotients listed in Table I of Martin (1996).
Table I of Martin (1996) for this q-series has exponent of 24 wrong. Number 54 should read 2^(-1)*4^4*6^(-1)*8^(-1)*12^4*24^(-1) (in column g).
The present expansion corresponds in Martin's notation to
1^(-1)*2^4*3^(-1)*4^(-1)*6^4*12^(-1). For the expansion of the (corrected) Nr. 54 of Martin's reference see A271231. One finds for the p-defects prime(m) - N(prime(m)) = A271230(m) of the elliptic curve y^2 = x^3 + x^2 + x (mod prime(m)), where N(prime(n)) = A271229(n) is the number of solutions, the modularity pattern A271231(prime(m)) = A271230(m), m >= 1. - Wolfdieter Lang, Apr 18 2016
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LINKS
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FORMULA
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Expansion of q^(-1/2) * eta(q^2)^4 * eta(q^6)^4 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)) in powers of q.
Expansion of f(x) * f(-x^2) * f(x^3) * f(-x^6) in powers of x where f() is a Ramanujan theta function.
Euler transform of period 12 sequence [ 1, -3, 2, -2, 1, -6, 1, -2, 2, -3, 1, -4, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 48 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: Product_{k>0} (1 - (-x)^k) * (1 - x^(2*k)) * (1 - (-x)^(3*k)) * (1 - x^(6*k)).
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EXAMPLE
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G.f. = 1 + x - 2*x^2 + x^4 - 4*x^5 - 2*x^6 - 2*x^7 + 2*x^8 + 4*x^9 + 8*x^11 - ...
G.f. = q + q^3 - 2*q^5 + q^9 - 4*q^11 - 2*q^13 - 2*q^15 + 2*q^17 + 4*q^19 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ x^2] QPochhammer[ -x^3] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Mar 31 2015 *)
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PROG
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(PARI) {a(n) = if(n<0, 0, ellak( ellinit([0, 1, 0, 1, 0], 1), 2*n + 1))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^6 + A)^4 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)), n))};
(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==3, 1, 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)))};
(PARI) q='q+O('q^220); Vec( (eta(q^2)*eta(q^6))^4 / (eta(q^1)*eta(q^3)*eta(q^4)*eta(q^12) ) ) \\ Joerg Arndt, Sep 12 2016
(Magma) A := Basis( CuspForms( Gamma0(48), 2), 147); A[1] + A[3]; /* Michael Somos, Mar 31 2015 */
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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