A138515 -id:A138515 - cp's OEIS frontend

A267859 The p-defect p - N(p) of the elliptic curve y^2 = x^3 + x for primes p congruent to 1 modulo 4 (A002144).

2, -6, 2, 10, 2, 10, -14, 10, -6, 10, 18, 2, -6, -14, -22, -14, -22, 26, 18, -14, 2, -30, 26, -30, 2, 26, 18, 10, 34, 26, -22, 18, 10, 34, -14, 34, -38, 2, -6, -30, 34, -14, 42, -38, 10, -22, 42, -38, 26, 2, -46, 10, 34, -38, 50, 26, 50, -46, 2, 10, -30, -54, 18, -38, 50, 34, -22, 10, 50, -54

Offset: 1

Wolfdieter Lang, Feb 06 2016

sign

See A002172 for a differently signed sequence.
The number N(p) of solutions modulo a prime p of the elliptic curve y^2 = x^3 + x (of discriminant -4) is given for all p in A095978.
The p-defect a_p = p - N(p) for prime 2 and primes congruent to 3 modulo 4 vanishes.
A002144(n) - (a(n)/2)^2  = (2*A002973(n))^2, n >= 1. See the formula for A095978 for primes 1 (mod 4).
This sequence gives also the non-vanishing p-defects of the elliptic curve y^2 = x^3 - 4*x. See a comment on A138515 with the Martin and Ono link for the modularity series for these two elliptic curves. - Wolfdieter Lang, May 26 2016

			n = 2: p = A002144(2) = 13 = A000040(6), m = 6, a(2) = 13 - A095978(6) = 13 - 19  = -6.
n = 2:  -6 = A138515((A002144(2) - 1)/4) =
A138515(3) = -6. - _Wolfdieter Lang_, May 26 2016

J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 398. In the 4th ed., 2014, p. 371.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000040, A002144, A002172, A095978, A138515.

terms = 100; A002144 = Select[Range[5, 20*terms, 4], PrimeQ]; A095978[n_] := Module[{p, xy, x}, p = Prime[n]; If[n==1 || Mod[p, 4]==3, Return[p]]; xy = {Re[#], Im[#]}& @ FactorInteger[p, GaussianIntegers -> True][[2, 1]]; x = SelectFirst[xy, OddQ]; If[Mod[x, 4]==1, p - 2*x, p + 2*x]]; a[n_] := (p = A002144[[n]]; m = PrimePi[p]; p - A095978[m]); Array[a, terms] (* Jean-François Alcover, Feb 26 2016, after Robert Israel (A095978) *)

a(n) = A002144(n) - A095978(m) with A002144(n) = A000040(m), n >= 1.

a(n) = A138515((A002144(n) - 1)/4), n >= 1. - Wolfdieter Lang, May 26 2016

A280339 Expansion of phi(x)^2 * chi(x^2)^4 * f(-x)^2 in powers of x where phi(), chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, -1, -2, -5, -14, 4, 12, 5, 40, 0, -26, 11, -68, -15, 30, -18, 106, 3, -50, -10, -182, 29, 104, 10, 270, 11, -130, 37, -360, -51, 164, -16, 506, -30, -266, -65, -686, 62, 320, 53, 898, 22, -414, 50, -1206, -61, 612, -52, 1560, -4, -696, -81, -1958, 120

Offset: 0

Michael Somos, Dec 31 2016

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 2*x - x^2 - 2*x^3 - 5*x^4 - 14*x^5 + 4*x^6 + 12*x^7 + 5*x^8 + ...
G.f. = q^-1 + 2*q^3 - q^7 - 2*q^11 - 5*q^15 - 14*q^19 + 4*q^23 + 12*q^27 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A138515, A279955.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^2 QPochhammer[ x]^2 QPochhammer[ -x^2, x^4]^4, {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^4 + A)^4 / (eta(x + A)^2 * eta(x^8 + A)^4), n))};

Expansion of phi(-x^4)^2 * chi(-x^4)^2 * f(x)^2 in powers of x where phi(), chi(), f() are Ramanujan theta functions.
Expansion of q^(1/4) * eta(q^2)^6 * eta(q^4)^4 / (eta(q)^2 * eta(q^8)^4) in powers of q.
Euler transform of period 8 sequence [2, -4, 2, -8, 2, -4, 2, -4, ...].
a(n) = (-1)^n * A279955(n).
a(3*n + 1) / a(1) == A138515(n) (mod 3). a(3^3*n + 7) / a(7) == A138515(n) (mod 3^2).

A127826 Coefficients of L-series for elliptic curve "256a1": y^2 = x^3 + x^2 - 3*x + 1.

Original entry on oeis.org

1, -2, 0, 0, 1, -6, 0, 0, -6, -2, 0, 0, -5, 4, 0, 0, 12, 0, 0, 0, 6, 10, 0, 0, -7, 12, 0, 0, 4, -6, 0, 0, 0, 14, 0, 0, -2, 10, 0, 0, -11, -18, 0, 0, -18, 0, 0, 0, 10, -6, 0, 0, 0, -6, 0, 0, 18, 0, 0, 0, 25, -12, 0, 0, -20, -18, 0, 0, 6, -22, 0, 0, 0, 14, 0, 0, -6, 0, 0, 0, 0, -2, 0, 0, -13, -2, 0, 0, 12, -18, 0, 0, 0, 36

Offset: 0

Michael Somos, Jan 30 2007

sign

			q - 2*q^3 + q^9 - 6*q^11 - 6*q^17 - 2*q^19 - 5*q^25 + 4*q^27 + 12*q^33 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
W. Stein, Modular Forms Database.

Convolution of A138515(q^4) and A112172.

f := qEigenform(EllipticCurve(CremonaDatabase(), "256a1"), 188); [ Coefficient(f, n) : n in [ k : k in [0..188] | IsOdd(k) ] ] ; /* Klaus Brockhaus, Feb 01 2007 */

eta[q_] := q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-1/2)* eta[q^8]^8/(eta[q^4]*eta[q^16])^2*(eta[q^4]/eta[q^16] - 2*eta[q^16] /eta[q^4]), {q, 0, 150}], q]; Table[a[[n]], {n, 1, 100}] (* G. C. Greubel, Jul 25 2018 *)

{a(n)=local(A, p, e, x, y); if(n<0, 0, n=2*n+1; A=factor(n); prod(k=1,matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 0, if(p%8>4, if(e%2, 0, (-p)^(e/2)), for(i=1, sqrtint(p\2), if(issquare(p-2*i^2, &x), y=i; break)); a0=1; a1=y=2*x*(-1)^((x%8>4)+(y%4>0)); for(i=2,e, x=y*a1-p*a0; a0=a1; a1=x); a1))))) }

{a(n)=local(A); if(n<0, 0, A=x*O(x^(n\4)); polcoeff( eta(x^2+A)^8/ eta(x+A)^2/ eta(x^4+A)^2* ((n%4==0)*eta(x+A)/eta(x^4+A) -(n%4==1)*2*eta(x^4+A)/eta(x+A)),n\4))}

{a(n) = ellak( ellinit([0, 1, 0, -3, 1], 1), 2*n + 1)}

a(n)=b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1+(-1)^e)/2*(-p^2)^(e/2) if p == 5,7 (mod 8), b(p^e) = b(p)*b(p^(e-1)) - p*b(p^(e-2)) if p == 1,3 (mod 8) and b(p) = 2*x*(-1)^((x mod 8 > 4) + (y mod 4) > 0) where p = x^2 + 2*y^2.
a(4n+2)=a(4n+3)=0.
Expansion of q^(-1/2) * eta(q^8)^8 / (eta(q^4) * eta(q^16))^2 * (eta(q^4) / eta(q^16) - 2 * eta(q^16) / eta(q^4)) in powers of q.
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).

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A267859 The p-defect p - N(p) of the elliptic curve y^2 = x^3 + x for primes p congruent to 1 modulo 4 (A002144).

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A280339 Expansion of phi(x)^2 * chi(x^2)^4 * f(-x)^2 in powers of x where phi(), chi(), f() are Ramanujan theta functions.

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A127826 Coefficients of L-series for elliptic curve "256a1": y^2 = x^3 + x^2 - 3*x + 1.

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