A140686 -id:A140686 - cp's OEIS frontend

A107498 Theta series of quadratic form with Gram matrix [ 4, -1, 1, 1; -1, 10, 3, 3; 1, 3, 10, -3; 1, 3, -3, 88].

1, 0, 2, 0, 0, 4, 4, 4, 6, 0, 0, 8, 0, 2, 0, 8, 0, 0, 6, 8, 12, 8, 0, 0, 12, 0, 6, 0, 8, 0, 0, 12, 14, 8, 8, 0, 0, 4, 0, 8, 0, 16, 0, 0, 24, 16, 16, 24, 0, 0, 26, 0, 14, 0, 36, 0, 0, 20, 24, 28, 44, 0, 0, 32, 0, 12, 0, 20, 0, 0, 40, 36, 58, 16, 0, 0, 52, 0, 24

Offset: 0

N. J. A. Sloane, May 28 2005

nonn

G.f. is theta_2 in the Parry 1979 reference on page 166. This theta series is an element of the space of modular forms on Gamma_0(169) of weight 2 and dimension 21. - Andy Huchala, May 13 2023

			G.f. = 1 + 2*q^4 + 4*q^10 + 4*q^12 + ...

Andy Huchala, Table of n, a(n) for n = 0..20000
W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170.

Cf. A107497 - A107505, A140686.

prec := 60;
ls := [[4, -1, 1, 1], [-1, 10, 3, 3], [1, 3, 10, -3], [1, 3, -3, 88]];
S := Matrix(ls);
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
T := ThetaSeries(L, 48);
coeffs := [Coefficients(T)[2*i-1] : i in [1..23]];
Coefficients(&+[coeffs[i]*B[i] :i in [1..13]]+&+[coeffs[i+1]*B[i] :i in [14..19]] + coeffs[22]*B[20] + coeffs[23]*B[21]); // Andy Huchala, May 13 2023

Name clarified and more terms from Andy Huchala, May 13 2023

A107497 Theta series of quadratic form with Gram matrix [ 2, 1, 1, 1; 1, 20, 7, 7; 1, 7, 20, 7; 1, 7, 7, 46].

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 8, 0, 8, 2, 4, 0, 10, 4, 0, 0, 0, 0, 12, 4, 0, 6, 6, 12, 0, 8, 20, 0, 0, 0, 0, 16, 22, 0, 24, 8, 24, 0, 32, 20, 0, 0, 0, 0, 36, 14, 0, 20, 14, 16, 0, 24, 32, 0, 0, 0, 0, 20, 28, 0, 30, 12, 44, 0, 24, 24, 0, 0, 0, 0, 28, 44, 0, 32

Offset: 0

N. J. A. Sloane, May 28 2005

nonn

G.f. is theta_1 in the Parry 1979 reference on page 166. This theta series is an element of the space of modular forms on Gamma_0(169) of weight 2 and dimension 21. - Andy Huchala, May 13 2023

			G.f. = 1 + 2*q^2 + 2*q^8 + 2*q^18 + ...

Andy Huchala, Table of n, a(n) for n = 0..20000
W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170.

Cf. A107498-A107505, A140686.

prec := 60;
ls := [[2,1,1,1],[1,20,7,7],[1,7,20,7],[1,7,7,46]];
S := Matrix(ls);
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
T := ThetaSeries(L,44);
coeffs := [Coefficients(T)[2*i-1] : i in [1..23]];
Coefficients(&+[coeffs[i]*B[i] :i in [1..13]]+&+[coeffs[i+1]*B[i] :i in [14..19]] + coeffs[22]*B[20] + coeffs[23]*B[21]); // Andy Huchala, May 13 2023

Name clarified and more terms from Andy Huchala, May 13 2023

A045399 Primes congruent to {0, 3, 5, 6} mod 7.

Original entry on oeis.org

3, 5, 7, 13, 17, 19, 31, 41, 47, 59, 61, 73, 83, 89, 97, 101, 103, 131, 139, 157, 167, 173, 181, 199, 223, 227, 229, 241, 251, 257, 269, 271, 283, 293, 307, 311, 313, 349, 353, 367, 383, 397, 409, 419, 433, 439

Offset: 1

N. J. A. Sloane

Primes p such that A140686(p) = 0. - Michael Somos, Feb 12 2011

Primes (along with 2) which cannot be written in the form a^2 + 7*b^2, where a > 0, b > 0. - V. Raman, Sep 08 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040.

[ p: p in PrimesUpTo(600) | p mod 7 in {0, 3, 5, 6} ]; // Vincenzo Librandi, Aug 12 2012

Select[Prime[Range[300]],MemberQ[{0,3,5,6},Mod[#,7]]&] (* Vincenzo Librandi, Aug 12 2012 *)

cp's OEIS Frontend

A107498 Theta series of quadratic form with Gram matrix [ 4, -1, 1, 1; -1, 10, 3, 3; 1, 3, 10, -3; 1, 3, -3, 88].

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A107497 Theta series of quadratic form with Gram matrix [ 2, 1, 1, 1; 1, 20, 7, 7; 1, 7, 20, 7; 1, 7, 7, 46].

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Extensions

A045399 Primes congruent to {0, 3, 5, 6} mod 7.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica