A159819 -id:A159819 - cp's OEIS frontend

A030188 Expansion of q^(-1/2) * eta(q) * eta(q^2) * eta(q^3) * eta(q^6) in powers of q.

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, 7, 14, -16, 2, -16, -2, 2, 0, 12, 8, 24, -9, -4, 6, 0, -4, 12, 6, 2, -12, 8, 0, 0

Offset: 0

N. J. A. Sloane

sign

Number 43 of the 74 eta-quotients listed in Table I of Martin (1996).

Newform number 1 of degree 1 in Full modular forms space of level 24, weight 2 and trivial character.

			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 + ...

J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 415. Exer. 47.3.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
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

Cf. A030209, A159819. A258090.

Basis( CuspForms( Gamma0(24), 2), 192) [1]; /* Michael Somos, May 27 2014 */

qEigenform( EllipticCurve( [0, -1, 0, 1, 0]), 191); /* Michael Somos, Jun 12 2014 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^2] QPochhammer[ x^3] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, May 17 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A), n))}; /* Michael Somos, Apr 02 2005 */

{a(n) = if( n<0, 0, n = 2*n + 1; ellak( ellinit([ 0, -1, 0, -4, 4], 1), n))}; /* Michael Somos, Apr 02 2005 */

{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)^e, a0 = 1; a1 = y = -sum( x=0, p-1, kronecker( x^3 - x^2 - 4*x + 4, p)); for( i=2, e, x = y*a1 - p*a0; a0 = a1; a1 = x); a1)))}; /* Michael Somos, Aug 13 2006 */

CuspForms( Gamma0(24), 2, prec=192).0 # Michael Somos, May 24 2013

Euler transform of period 6 sequence [ -1, -2, -2, -2, -1, -4, ...]. - Michael Somos, Apr 02 2005
Given g.f. A(x), then B(x) = x * A(x)^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = u^2 * v * w + 4 * u * v^2 * w + 16 * u * v * w^2 + 4 * u^2 * w^2 - v^4. - Michael Somos, Apr 02 2005
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = (-1)^e, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) otherwise where b(p) = p+1 - number of solutions to y^2 = x^3 - x^2 - 4*x + 4 modulo p including the point at infinity. - Michael Somos, Mar 04 2011
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(6*k)).
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 24 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 08 2007
Coefficients of L-series for elliptic curve "24a1": y^2 = x^3 - x^2 - 4*x + 4. - Michael Somos, Apr 02 2005
a(n) = (-1)^n * A159819(n). a(3*n + 1) = -a(n). Convolution square is A030209. - Michael Somos, Mar 13 2012
a(3*n + 2) = -2 * A258090(n). - Michael Somos, May 19 2015

A271231 Expansion of the modular cusp form ( eta(q^4) * eta(q^12) )^4 / ( eta(q^2) * eta(q^6) * eta(q^8) * eta(q^24) ), where eta is Dedekind's eta function.

Original entry on oeis.org

0, 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, 4, 0, -4, 0, 6, 0, -6, 0, 0, 0, -8, 0, -8, 0, 2, 0, -4, 0, -18, 0, -16

Offset: 0

Wolfdieter Lang, Apr 19 2016

The modularity pattern of the elliptic curve y^2 = x^3 + x^2 + x considered modulo prime(m) is seen from a(prime(m)) = prime(m) - N(prime(m)) = A271230(m), where N(prime(m))= A271229(m) is the number of solutions of this congruence. That is, the p-defect coincides with the prime indexed expansion coefficient (here for all primes).
This modular cusp form of weight 2 and level N = 48 = 2^4*3 is Nr. 54 in Martin's Table 1 (corrected by giving the 24 the missing exponent -1). See also the Michael Somos link where this correction has been observed.
This modular cusp form is a simultaneous eigenform of every Hecke operators T_p, with p a prime not 2 or 3 (bad primes) with eigenvalue lambda(p) = a(p). (See the Martin reference, Proposition 33, p. 4851.)
In the Martin and Ono reference, p. 3173 (Theorem 2), this cusp form appears (in the corrected version) in the row Conductor 48, and it is there related to the elliptic curve y^2 = x^3 + x^2 - 4*x - 4. The p-defects of this curve coincide with the ones of the curve  y^2 = x^3 + x^2 + x modulo primes p given in A271230. - Wolfdieter Lang, Apr 21 2016
Multiplicative. See A159819 for formula. - Andrew Howroyd, Aug 06 2018

			n=2: a(2) = A271230(1) = 0.
n=5: a(5) = A271230(3) = -2.
See the example section of A271229 for the solutions for the first primes.

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.
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

Cf. A159819, A271229, A271230.

QP = QPochhammer;
a[n_] := If[OddQ[n], SeriesCoefficient[QP[-x] QP[x^2] QP[-x^3] QP[x^6], {x, 0, (n-1)/2}], 0];
a /@ Range[0, 100] (* Jean-François Alcover, Sep 19 2019 *)

q='q+O('q^220); concat([0], Vec( (eta(q^4)*eta(q^12))^4 / (eta(q^2)*eta(q^6)*eta(q^8)*eta(q^24) ) ) ) \\ Joerg Arndt, Sep 12 2016

a(2*n+1) = A159819(n), a(2*n) = 0.
O.g.f.: Expansion in q = exp(2*Pi*i*z) with Im(z) > 0 of (eta(4*z)*eta(12*z))^4 / (eta(2*z)*eta(6*z)*eta(8*z)*eta(24*z)), where eta(z) = q^(1/24)*Product_{n >= 1} (1 - q^n) is the Dedekind function with q = q(z) given above, and i is the imaginary unit.
a(prime(m)) = A271230(m), m >= 1.

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.

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A030188 Expansion of q^(-1/2) * eta(q) * eta(q^2) * eta(q^3) * eta(q^6) in powers of q.

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A271231 Expansion of the modular cusp form ( eta(q^4) * eta(q^12) )^4 / ( eta(q^2) * eta(q^6) * eta(q^8) * eta(q^24) ), where eta is Dedekind's eta function.

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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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