A095978 -id:A095978 - cp's OEIS frontend

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

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, -6, -14, 18, -11, -12, 0, 0, -22, 0, 20, -14, -6, -22, 0, 0, 23, 26, 0, 18, 4, 0, -14, 2, 0, 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

Offset: 0

Michael Somos, Mar 22 2008

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 58 of the 74 eta-quotients listed in Table I of Martin (1996). - Michael Somos, Mar 16 2012
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

			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 + ...
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 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)
LMFDB, Elliptic Curve 64.a4 (Cremona label 64a4)
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002171, A095978, A138514, A267859, A280339.

A := Basis( CuspForms( Gamma0(64), 2), 342); A[1] + 2*A[3]; /* Michael Somos, May 15 2015 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] QPochhammer[ -q])^2, {q, 0, n}]; (* Michael Somos, May 15 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^4 / (QPochhammer[ q] QPochhammer[ q^4]))^2, {q, 0, n}]; (* Michael Somos, May 15 2015 *)

{a(n) = ellak( ellinit( [ 0, 0, 0, 1, 0], 1), 4*n + 1)};

{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))};

{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)))))};

Coefficients of L-series for elliptic curve "64a4": y^2 = x^3 + x.
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.
Euler transform of period 4 sequence [2, -6, 2, -4, ...].
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).
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.
G.f.: (Product_{k>0} (1 - x^(2*k))^2 * (1 + x^(2*k - 1)))^2.
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.
G.f. for{b(n)}:
  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

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).

Original entry on oeis.org

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

A276730 Number of solutions to y^2 == x^3 + 4*x (mod p) as p runs through the primes.

Original entry on oeis.org

2, 3, 7, 7, 11, 7, 15, 19, 23, 39, 31, 39, 31, 43, 47, 39, 59, 71, 67, 71, 79, 79, 83, 79, 79, 103, 103, 107, 103, 127, 127, 131, 159, 139, 135, 151, 135, 163, 167, 199, 179, 199, 191, 207, 199, 199, 211, 223, 227, 199, 207, 239, 271, 251, 255, 263, 295, 271, 295, 271

Offset: 1

Seiichi Manyama, Sep 16 2016

nonn

This elliptic curve corresponds to a weight 2 newform which is an eta-quotient, namely, (eta(4t)*eta(8t))^2, see Theorem 2 in Martin & Ono.
It appears that a(n) = prime(n) iff prime(n) == 2 or 3 (mod 4). - Robert Israel, Sep 28 2016 This is true due to the L-function of this elliptic curve. See A278720. - Wolfdieter Lang, Dec 22 2016
The rational solutions of y^2 = x^3  + 4*x are (x,y) = (0,0), (2,4), (2,-4). See the Keith Conrad link, Corollary 3.17., p. 9. - Wolfdieter Lang, Dec 01 2016
For the p-defects p - N(p) see A278720. - Wolfdieter Lang, Dec 22 2016

			The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used.
The solutions (x, y) of y^2 == x^3 + 4*x (mod prime(n)) begin:
n, prime(n), a(n)\  solutions (x, y)
1,   2,       2:   (0, 0), (1, 1)
2,   3,       3:   (0, 0), (2, 1), (2, 2)
3,   5,       7:   (0, 0), (1, 0), (2, 1),
                   (2, 4), (3, 2), (3, 3),
                   (4, 0)
4,   7,       7:   (0, 0), (2, 3), (2, 4),
                   (3, 2), (3, 5), (6, 3),
                   (6, 4)
...
The solutions (x, y) of y^2 == x^3 - x (mod prime(n)) begin:
n, prime(n), a(n)\  solutions (x, y)
1,   2,       2:   (0, 0), (1, 0);
2,   3,       3:   (0, 0), (1, 0), (2, 0);
3,   5,       7:   (0, 0), (1, 0), (2, 1),
                   (2, 4), (3, 2), (3, 3),
                   (4, 0);
4,   7,       7:   (0, 0), (1, 0), (4, 2),
                   (4, 5), (5, 1), (5, 6),
                   (6, 0);
... - _Wolfdieter Lang_, Dec 22 2016

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Keith Conrad, Expository papers, Proofs by Descent.
Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.

Cf. A095978, A272207, A278720.

seq(nops([msolve(y^2-x^3-4*x, ithprime(n))]),n=1..100); # Robert Israel, Sep 28 2016

require 'prime'
def A(a3, a2, a4, a6, n)
  ary = []
  Prime.take(n).each{|p|
    a = Array.new(p, 0)
    (0..p - 1).each{|i| a[(i * i + a3 * i) % p] += 1}
    ary << (0..p - 1).inject(0){|s, i| s + a[(i * i * i + a2 * i * i + a4 * i + a6) % p]}
  }
  ary
end
def A276730(n)
  A(0, 0, 4, 0, n)
end

a(n) is the number of solutions of the congruence y^2 == x^3 + 4*x (mod prime(n)), n >= 1.
a(n) is also the number
of solutions of the congruence y^2 == x^3 - x (mod prime(n)), n >= 1. - Wolfdieter Lang, Dec 22 2016 (See the Cremona link given in A278720).

A267858 Positions of entries of A002972 that are congruent to 1 modulo 4.

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 10, 11, 12, 18, 19, 21, 23, 25, 26, 27, 28, 29, 30, 32, 33, 34, 36, 38, 41, 43, 45, 47, 49, 50, 52, 53, 55, 56, 57, 59, 60, 63, 65, 66, 68, 69, 72, 73, 74, 77, 78, 85, 87, 88, 89, 90, 91, 93, 94, 95, 96, 100, 104, 105, 106, 108, 110, 112, 115, 119, 120, 122, 127, 128, 131

Offset: 1

Wolfdieter Lang, Feb 06 2016

nonn

This sequence is needed for the number of solutions modulo primes congruent to 1 modulo 4 of the elliptic curve y^2 = x^3 + x See A095978.

If a positive integer m is not in this sequence then A002972(m) == 3 (mod 4).

			n=1: A002972(1) = 1 == 1 (mod 4). But because m = 2 is not in this sequence A002972(2) = 3 == 3 (mod 4).

Cf. A002145, A002972.

pmax = 2000; odd[p_] := Module[{k, m}, 2m+1 /. ToRules[Reduce[k>0 && m >= 0 && (2k)^2 + (2m+1)^2 == p, {k, m}, Integers]]]; Reap[For[n=1; p=5, p < pmax, p = NextPrime[p], If[Mod[p, 4]==1, If[Mod[odd[p], 4]==1, Sow[n]]; n++]]][[2, 1]] (* Jean-François Alcover, Feb 26 2016 *)

A002972(a(n)) == 1 (mod 4), n >= 1.

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

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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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A276730 Number of solutions to y^2 == x^3 + 4*x (mod p) as p runs through the primes.

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A267858 Positions of entries of A002972 that are congruent to 1 modulo 4.

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