๐ŸŒฟ Sagebrush Atlas
Atlas โ€บ Modular forms โ€บ Weight 2 โ€บ Level 780 โ€บ 780.2.a

Newspace 780.2.a: S2new(ฮ“0(780))S_{2}^{\mathrm{new}}(\Gamma_0(780))

Newforms

LabelDimCoefficient fieldTrace formAL signsSignCMElliptic curve
780.2.a.a1Q\mathbb{Q} qโˆ’q3+q5โˆ’2q7+O(q8)q - q^{3} + q^{5} - 2q^{7} + O(q^{8})+ โˆ’ + 780a1 rank 1
780.2.a.b1Q\mathbb{Q} qโˆ’q3+q5+3q7+O(q8)q - q^{3} + q^{5} + 3q^{7} + O(q^{8})+ โˆ’ โˆ’ 780b1 rank 0
780.2.a.c1Q\mathbb{Q} q+q3โˆ’q5โˆ’2q7+O(q8)q + q^{3} - q^{5} - 2q^{7} + O(q^{8})โˆ’ + + 780c1 rank 1
780.2.a.d1Q\mathbb{Q} q+q3โˆ’q5โˆ’q7+O(q8)q + q^{3} - q^{5} - q^{7} + O(q^{8})โˆ’ + โˆ’ 780d1 rank 0
780.2.a.e2Q(73)\mathbb{Q}(\sqrt{73}) 2qโˆ’2q3โˆ’2q5โˆ’q7+O(q8)2q - 2q^{3} - 2q^{5} - q^{7} + O(q^{8})+ + +
780.2.a.f2Q(17)\mathbb{Q}(\sqrt{17}) 2q+2q3+2q5+3q7+O(q8)2q + 2q^{3} + 2q^{5} + 3q^{7} + O(q^{8})โˆ’ โˆ’ +

Hecke characteristic polynomials

The eigenvalue of T=T7+3T11+5T19+7T23+9T29+11T31T = T_{7} + 3T_{11} + 5T_{19} + 7T_{23} + 9T_{29} + 11T_{31} generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over Q\mathbb{Q} as one irreducible factor per orbit:

OrbitCharacteristic polynomial of TT
780.2.a.ax+50x + 50
780.2.a.bxโˆ’87x - 87
780.2.a.cx+62x + 62
780.2.a.dxโˆ’115x - 115
780.2.a.ex2โˆ’265x+16078x^{2} - 265x + 16078
780.2.a.fx2+11x+26x^{2} + 11x + 26

Dimensions

Newforms S2new(ฮ“0(780))S_{2}^{\mathrm{new}}(\Gamma_0(780))8
Old forms149
Cusp forms S2(ฮ“0(780))S_{2}(\Gamma_0(780))157
Eisenstein series E2(ฮ“0(780))E_{2}(\Gamma_0(780))23
Modular forms M2(ฮ“0(780))M_{2}(\Gamma_0(780))180

Decomposition of the old subspace

S2old(ฮ“0(780))=S_{2}^{\mathrm{old}}(\Gamma_0(780)) = S2new(ฮ“0(15))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 6} โŠ•\oplus S2new(ฮ“0(20))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(26))โŠ•8S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 8} โŠ•\oplus S2new(ฮ“0(30))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(39))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 6} โŠ•\oplus S2new(ฮ“0(52))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(52))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(65))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(65))^{\oplus 6} โŠ•\oplus S2new(ฮ“0(78))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(78))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(130))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(130))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(156))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(156))^{\oplus 2} โŠ•\oplus S2new(ฮ“0(195))โŠ•3S_{2}^{\mathrm{new}}(\Gamma_0(195))^{\oplus 3} โŠ•\oplus S2new(ฮ“0(260))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(260))^{\oplus 2} โŠ•\oplus S2new(ฮ“0(390))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(390))^{\oplus 2}

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.892 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of TT over Z\mathbb{Z} is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks
  • dim M^+ at levels [1, 2, 3, 4, 5, 6, 10, 12, 13, 15, 20, 26, 30, 39, 52, 60, 65, 78, 130, 156, 195, 260, 390, 780]: [0, 1, 1, 2, 1, 3, 3, 5, 1, 4, 6, 5, 10, 6, 10, 18, 8, 18, 24, 34, 32, 48, 92, 180] (certified at ell = 2147483629)
  • T = [(7, 1), (11, 3), (19, 5), (23, 7), (29, 9), (31, 11)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 3 primes, 0 rejected
Open in the notebook ยท JSON ยท stored data
Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(780, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 780:2 1000

โ† 779.2.a ยท 781.2.a โ†’