A363147 Primes q == 1 (mod 4) such that there is at least one equivalence class of quaternary quadratic forms of discriminant q not representing 2.

193, 233, 241, 257, 277, 281, 313, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593, 601, 613, 617, 641, 653, 661, 673, 677, 701, 709, 733, 757, 761, 769, 773, 797, 809, 821, 829, 853, 857, 877, 881, 929, 937

Offset: 1

Author

Andy Huchala, May 18 2023

Keywords

• nonn

Links

• Andy Huchala, Table of n, a(n) for n = 1..20000
• F. Hirzebruch, Modulflächen und Modulkurven zur symmetrischen Hilbertschen Modulgruppe, Annales scientifiques de l’É.N.S. 4e série, tome 11, no 1 (1978), p. 101-165. See page 135.
• Jürg Kramer, On the linear independence of certain theta-series, Mathematische Annalen 281.2 (1988): 219-228. See page 226.

Crossrefs

Cf. A307250, A363148.

Programs

Sage

bound = 100
P = Primes()
p = 2
for i in range(bound):
    p = P.next(p)
    if p % 4 == 1:
        K1. = NumberField(x^2 - p)
        K2. = NumberField(x^2 + p)
        K3. = NumberField(x^2 + 3*p)
        zeta = K1.zeta_function()
        h2 = len(K2.class_group())
        h3 = len(K3.class_group())
        H_plus = int(abs(.49+1/2*zeta(-1)+1/8 * h2 + 1/6*h3))
        H = (H_plus+int((p + 19)/24))/2
        if H_plus-H>0:
            print(p)