A271231 -id:A271231 - cp's OEIS frontend

A159819 Coefficients of L-series for elliptic curve "48a4": y^2 = x^3 + x^2 + x.

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

Offset: 0

Michael Somos, Apr 22 2009

sign

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

			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 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
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. A030188, A271229, A271230, A271231.

A := Basis( CuspForms( Gamma0(48), 2), 147); A[1] + A[3]; /* Michael Somos, Mar 31 2015 */

a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ x^2] QPochhammer[ -x^3] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Mar 31 2015 *)

{a(n) = if(n<0, 0, ellak( ellinit([0, 1, 0, 1, 0], 1), 2*n + 1))};

{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))};

{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)))};

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

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)).
a(n) = (-1)^n * A030188(n).

A271229 Number of solutions of the congruence y^2 == x^3 + x^2 + x (mod p) as p runs through the primes.

Original entry on oeis.org

2, 2, 7, 7, 15, 15, 15, 15, 15, 23, 39, 31, 47, 47, 47, 55, 63, 63, 63, 79, 63, 71, 79, 95, 95, 119, 119, 95, 111, 95, 119, 127, 143, 127, 135, 135, 159, 175, 191, 167, 191, 175, 191, 191, 215, 215, 191, 215, 239, 207, 223, 223, 223, 271, 255, 255, 279, 279, 303, 255

Offset: 1

Wolfdieter Lang, Apr 18 2016

The discriminant of the elliptic curve y^2 = x^3 + x^2 + x is -3.

			Here P(n) stands for prime(n).
n,  P(n), a(n)\ Solutions (x, y) modulo P(n)
1,   2,    2:  (0, 0), (1, 1)
2,   3:    2:  (0, 0), (1, 0)
3,   5,    7:  (0, 0), (2, 2), (2, 3), (3, 2), (3, 3),
               (4, 2), (4, 3)
4,   7,    7:  (0, 0), (2, 0), (3, 2), (3, 5), (4, 0),
               (5, 1), (5, 6)
5,  11,   15:  (0, 0), (1, 5), (1, 6), (2, 5), (2, 6),
               (5, 1), (5, 10), (6, 4), (6, 7), (7, 5),
               (7, 6), (8, 1), (8, 10), (9, 4), (9, 7)
...
-------------------------------------------------------

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A159819. A271230. A271231.

a(n) is the number of solutions of the congruence y^2 == x^3 + x^2 + x (mod prime(n)), n >= 1.

a(n) = prime(n) - A271230(n), n >= 1.

A271230 P-defects p - N(p) of the congruence y^2 == x^3 + x^2 + x (mod p) for primes p, where N(p) is the number of solutions.

Original entry on oeis.org

0, 1, -2, 0, -4, -2, 2, 4, 8, 6, -8, 6, -6, -4, 0, -2, -4, -2, 4, -8, 10, 8, 4, -6, 2, -18, -16, 12, -2, 18, 8, 4, -6, 12, 14, 16, -2, -12, -24, 6, -12, 6, 0, 2, -18, -16, 20, 8, -12, 22, 10, 16, 18, -20, 2, 8, -10, -8, -26, 26

Offset: 1

Wolfdieter Lang, Apr 18 2016

The modularity pattern series is the expansion of the (corrected) Nr. 54 modular cusp form of weight 2 and level N=48 given in the table 1 of the Martin reference, i.e., (eta(4*z) * eta(12*z)^4 / (eta(2*z) * eta(6*z) * eta(8*z) * eta(24*z)) in powers of q = exp(2*Pi*i*z), with Im(z) > 0, where i is the imaginary unit. Here eta(z) = q^{1/24}*Product_{n>=1} (1-q^n) is the Dedekind eta function. See A271231 for this expansion. Note that also for the possibly bad prime 2 and the bad prime 3 (the discriminant of this elliptic curve is -3) this expansion gives the correct p-defect.

The identical p-defects occur for the elliptic curve y^2 = x^3 + x^2 - 4*x - 4 taken modulo prime(n). See the Martin and Ono reference, p. 3173, row Conductor 48, and A271231 (checked up to prime(100) = 541). - Wolfdieter Lang, Apr 21 2016

			See the example section of A271229.
n = 3, prime(3) = 5, A271229(5) = 7, a(3) = 5 - 7 = -2.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
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
Haode Yan, Yongbo Xia, Chunlei Li, Tor Helleseth, Maosheng Xiong and Jinquan Luo, The Differential Spectrum of the Power Mapping x^(p^n-3), arXiv:2108.03088 [cs.IT], 2021. See Table II p. 7.

Cf. A159819, A271229, A271231.

a(n) = prime(n) - A271229(n), n >= 1, where A271229(n) is the number of solutions of the congruence y^2 == x^3 + x^2 + x (mod prime(n)).

a(n) = A271231(prime(n)), n >=1.

A276649 Primes such that A271229(n) = prime(n).

Original entry on oeis.org

2, 7, 47, 191, 383, 439, 1151, 1399, 2351, 2879, 3119, 3511, 3559, 4127, 5087, 5431, 6911, 8887, 9127, 9791, 9887, 12391, 13151, 14407, 15551, 16607, 19543, 20399, 21031, 21319, 21839, 23039, 25391, 26399, 28087, 28463, 28711, 29287, 33223, 39551, 43103, 44879

Offset: 1

Seiichi Manyama, Sep 11 2016

nonn

These terms are the primes such that A271231(p) == 0 (mod p).
These terms are the primes prime(A271230(n)) such that A271230(n) = 0.
How is this related to A167860? - R. J. Mathar, May 16 2023

Cf. A271229, A271230, A271231, A167860.

lista(nn) = {q = 'q+O('q^(nn+1)); ser = q*(eta(q^4)*eta(q^12))^4 / (eta(q^2)*eta(q^6)*eta(q^8)*eta(q^24)); forprime (p=1, nn, c = polcoeff(ser, p); if ((c % p) == 0, print1(p, ", ")););} \\ Michel Marcus, Sep 14 2016

A276847 Expansion of eta(q^2) * eta(q^4) * eta(q^6) * eta(q^12) in powers of q.

Original entry on oeis.org

1, 0, -1, 0, -2, 0, 0, 0, 1, 0, 4, 0, -2, 0, 2, 0, 2, 0, -4, 0, 0, 0, -8, 0, -1, 0, -1, 0, 6, 0, 8, 0, -4, 0, 0, 0, 6, 0, 2, 0, -6, 0, 4, 0, -2, 0, 0, 0, -7, 0, -2, 0, -2, 0, -8, 0, 4, 0, 4, 0, -2, 0, 0, 0, 4, 0, -4, 0, 8, 0, 8, 0, 10, 0, 1, 0, 0, 0, -8, 0, 1, 0

Offset: 1

Seiichi Manyama, Sep 22 2016

The bisection of this sequence containing all nonzero terms is A030188.

Multiplicative. See A030188 for formula. - Andrew Howroyd, Jul 31 2018

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

Cf. A030188, A271231, A276807, A276649.

CoefficientList[Series[QPochhammer[x^2] QPochhammer[x^4] QPochhammer[x^6] QPochhammer[x^12], {x, 0, 100}], x] (* Jan Mangaldan, Jan 04 2017 *)

a(4*n-3) = A271231(4*n-3), a(4*n-2) = 0, a(4*n-1) = -A271231(4*n-1), a(4*n) = 0.
G.f.: x * Product_{k>0} (1 - x^(2*k)) * (1 - x^(4*k)) * (1 - x^(6*k)) * (1 - x^(12*k)).
a(2*n+1) = A030188(n). - Michel Marcus, Sep 25 2016
Euler transform of period 12 sequence [0, -1, 0, -2, 0, -2, 0, -2, 0, -1, 0, -4, ...]. - Georg Fischer, Nov 17 2022

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A159819 Coefficients of L-series for elliptic curve "48a4": y^2 = x^3 + x^2 + x.

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A271229 Number of solutions of the congruence y^2 == x^3 + x^2 + x (mod p) as p runs through the primes.

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A271230 P-defects p - N(p) of the congruence y^2 == x^3 + x^2 + x (mod p) for primes p, where N(p) is the number of solutions.

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A276649 Primes such that A271229(n) = prime(n).

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A276847 Expansion of eta(q^2) * eta(q^4) * eta(q^6) * eta(q^12) in powers of q.

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