A273163 -id:A273163 - cp's OEIS frontend

A030205 Expansion of q^(-1/2) * eta(q)^2 * eta(q^5)^2 in power of q.

1, -2, -1, 2, 1, 0, 2, 2, -6, -4, -4, 6, 1, 4, 6, -4, 0, -2, 2, -4, 6, -10, -1, -6, -3, 12, -6, 0, 8, 12, 2, 2, -2, 2, -12, -12, 2, -2, 0, 8, -11, 6, 6, -12, -6, 4, 8, 4, 2, 0, 6, 14, 4, -6, 2, -4, -6, -6, 2, -12, -11, -12, -1, 2, 20, 0, -8, -4, 18, -4, 12, 0

Offset: 0

N. J. A. Sloane

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This eta-quotient of conductor 20 is one of the twelve weight 2 newforms listed by Martin and Ono.
The associated elliptic curve is "20a1": y^2 = x^3 + x^2 + 4*x + 4 or "20a2": y^2 = x^3 + x^2 - x.
Number 39 of the 74 eta-quotients listed in Table I of Martin (1996).
The mentioned eta-quotient is in fact eta^2(2*z) * eta^2(10*z) with q = exp(2*Pi*i*tau) with Im(tau) > 0, I^2 = -1, with the q-expansion coefficients b(n) from the Michael Somos Oct Aug 13 2006 formula: b(2*n) = 0 and b(2*n+1) = a(n), for n >= 0. A273163(k) = b(prime(k)), k >= 1. See also the comments on multiplicativity of b(n) (called there c(n)) with b(2^k) = b(2)^k = 0, b(5^k) = b(5)^k = (-1)^k, and b(prime(n)^k) = (sqrt(prime(n)))^k*S(k,A273163(n)/sqrt(prime(n))) with Chebyshev's S polynomials (A049310), for n = 2, and n >= 4 and k >= 2. Compare this with the b(p^(e+2)) recurrence given by Michael Somos, Oct 31 2005. - Wolfdieter Lang, May 23 2016

			G.f. = 1 - 2*x - x^2 + 2*x^3 + x^4 + 2*x^6 + 2*x^7 - 6*x^8 - 4*x^9 - 4*x^10 + ...
G.f. of b(n) from eta^2(2*z)*eta^2(10*z) = q - 2*q^3 - q^5 + 2*q^7 + q^9 + 2*q^13 + 2*q^15 - 6*q^17 - 4*q^19 + ..., where q = exp(2*Pi*I*z).

Seiichi Manyama, Table of n, a(n) for n = 0..1000
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
LMFDB, Space of modular forms of level 20 and weight 2.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Y. Martin and K. Ono, Eta-quotients and elliptic curves, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3169-3176. MR1401749 (97m:11057)
K. Ono, Ramanujan, taxicabs, birthdates, ZIP codes and twists, Amer. Math. Monthly, 104 (1997), 912-917. MR1490909 (98i:11020)
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Cf. A030210, A049310, A159817, A273163.

Basis( CuspForms( Gamma0(20), 2), 145) [1]; /* Michael Somos, May 27 2014 */

A := Basis( CuspForms( Gamma1(20), 2), 145); A[1] - 2*A[3]; /* Michael Somos, May 17 2015 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^5])^2, {x, 0, n}]; (* Michael Somos, May 28 2013 *)

{a(n) = if( n<0, 0, n = 2*n + 1; ellan( ellinit( [0, 1, 0, 4, 4], 1), n)[n])}; /* Michael Somos, Oct 31 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^5 + A))^2, n))}; /* Michael Somos, Oct 31 2005 */

{a(n) = if( n<0, 0, n = 2*n + 1; ellan( ellinit( [0, 1, 0, -1, 0], 1), n)[n])}; /* Michael Somos, Aug 13 2006 */

{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==5, (-1)^e, 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)))}; /* Michael Somos, Aug 13 2006 */

CuspForms( Gamma0(20), 2, prec=92).0; # Michael Somos, May 28 2013

Euler transform of period 5 sequence [ -2, -2, -2, -2, -4, ...]. - Michael Somos, Oct 31 2005
Given g.f. A(x), then B(x) = q * A(q)^2 satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u*w * (u + 8*v + 16*w) - v^3. - Michael Somos, Oct 31 2005
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(5^e) = (-1)^e, else b(p^(e+2)) = b(p)*b(p^(e+1)) - p*b(p^e). - Michael Somos, Oct 31 2005
a(n) = b(2*n + 1) and b(p) = p minus number of points of elliptic curve "20a1" or "20a2" modulo p. - Michael Somos, Aug 13 2006
G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(5*k)))^2.
a(121*n + 60) = -11 * a(n).
Convolution square is A030210. - Michael Somos, Jun 13 2014
a(n) = (-1)^n * A159817(n). - Michael Somos, Jun 10 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 20 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 10 2015

A272207 Number of solutions to the congruence y^2 == x^3 + x^2 + 4*x + 4 (mod p) as p runs through the primes.

Original entry on oeis.org

2, 5, 6, 5, 11, 11, 23, 23, 17, 23, 35, 35, 35, 53, 53, 59, 47, 59, 65, 83, 71, 71, 77, 95, 95, 95, 89, 113, 107, 119, 125, 131, 119, 143, 155, 131, 179, 173, 149, 179, 191, 191, 203, 167, 179, 191, 227, 233, 233, 215, 239, 263, 227, 251, 263, 281, 251, 251, 251, 275

Offset: 1

Wolfdieter Lang, May 20 2016

In the Martin and Ono reference, in Theorem 2, this elliptic curve appears in the fourth row, starting with conductor 20, as a strong Weil curve for the weight 2 newform (eta(2*z)*eta(10*z))^2, symbolically 2^2 10^2, with Im(z) > 0, and the Dedekind eta function. See A030205 which gives the q-expansion (q = exp(2*Pi*i*z)) of exp(-Pi*i*z)*(eta(z)*eta(5*z))^2. For the q-expansion of (eta(2*z)*eta(10*z))^2 one has interspersed 0's: 0, 1, 0, -2, 0, -1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 2, 0, -6, ... This modular cusp form of weight 2 appears as the 39th entry in Martin's Table I.
For the p-defect prime(n) - a(n) see A273163(n), n >= 1.
The discriminant of this elliptic curve is -400 = -2^4*5^2 (bad primes 2 and 5, also the prime divisors of the conductor).
The congruence y^2 == x^3 + x^2 - x has the same number of solutions modulo prime(n). See a comment on A030205. The discriminant equals +5.

			The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used.
The solutions (x, y) of y^2 == x^3 + x^2 + 4*x + 4 (mod prime(n)) begin:
n, prime(n), a(n)\  solutions (x, y)
1,   2,       2:   (0, 0), (1, 0)
2,   3,       5:   (0, 1), (0, 2), (1, 1),
                   (1, 2) (2, 0)
3,   5,       6:   (0, 2), (0, 3), (1, 0),
                   (2, 2), (2, 3), (4, 0)
4,   7,       5:   (0, 2), (0, 5), (4, 3),
                   (4, 4), (6, 0)
5,  11,      11:   (0, 2), (0, 9), (4, 1),
                   (4, 10), (5, 3), (5, 8),
                   (6, 4), (6, 7), (9, 5),
                   (9, 6), (10, 0)
...
The solutions (x, y) of y^2 == x^3 + x^2 - x (mod prime(n)) begin:
n, prime(n), a(n)\  solutions (x, y)
1,   2,       2:   (0, 0), (1, 1)
2,   3,       5:   (0, 0), (1, 1), (1, 2),
                   (2, 1) (2, 2)
3,   5,       6:   (0, 0), (1, 1), (1, 4),
                   (2, 0), (4, 1), (4, 4)
4,   7,       5:   (0, 0), (1, 1), (1, 6),
                   (6, 1), (6, 6)
5,  11,      11:   (0, 0), (1, 1), (1, 10),
                   (3, 0), (6, 2), (6, 9),
                   (7, 0), (9, 3), (9, 8),
                   (10, 1), (10, 10)
...

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.

Cf. A000040, A030205, A273163.

a(n) gives the number of solutions of the congruence y^2 == x^3 + x^2 + 4*x + 4 (mod prime(n)), n >= 1.

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

A276695 P-defects p - N(p) of the congruence y^2 == x^3 - x^2 + 4*x - 4 (mod p) for primes p, where N(p) is the number of solutions given for p = prime(n) by A276664(n).

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Sep 14 2016

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

Cf. A273163, A276664.

a(n) = prime(n) - A276664(n), n >= 1, where A276664(n) is the number of solutions to the congruence y^2 == x^3 - x^2 + 4*x - 4 (mod prime(n)).

If prime(n) == 1 (mod 4), a(n) = A273163(n). If prime(n) == 3 (mod 4), a(n) = -A273163(n).

A276030 Primes p such that A272207(p) = p.

Original entry on oeis.org

2, 11, 131, 251, 491, 599, 1439, 3371, 5639, 5879, 6971, 7079, 8039, 8291, 9839, 10799, 11171, 12119, 14879, 16931, 17159, 18839, 23039, 23159, 25919, 50291, 53411, 53639, 59051, 69371, 74771, 74891, 75239, 81119, 81359, 117839, 119039, 126839, 130811, 131771

Offset: 1

Seiichi Manyama, Sep 10 2016

nonn

These terms are the primes prime(A273163(n)) for which A273163(n) = 0.
These terms are the primes for which A276491(p) == 0 (mod p).
These terms are the primes p = prime(n) for which A276664(n) = p.
These terms are the primes prime(A276695(n)) for which A276695(n) = 0.

			2 = A272207(1) = prime(1),
11 = A272207(5) = prime(5),
131 = A272207(32) = prime(32),
251 = A272207(54) = prime(54).

Cf. A272207, A273163, A276491, A276664, A276695.

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A030205 Expansion of q^(-1/2) * eta(q)^2 * eta(q^5)^2 in power of q.

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A272207 Number of solutions to the congruence y^2 == x^3 + x^2 + 4*x + 4 (mod p) as p runs through the primes.

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A276695 P-defects p - N(p) of the congruence y^2 == x^3 - x^2 + 4*x - 4 (mod p) for primes p, where N(p) is the number of solutions given for p = prime(n) by A276664(n).

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A276030 Primes p such that A272207(p) = p.

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