๐ŸŒฟ Sagebrush Atlas
Atlas โ€บ Modular forms โ€บ Weight 4 โ€บ Level 240 โ€บ 240.4.a

Newspace 240.4.a: S4new(ฮ“0(240))S_{4}^{\mathrm{new}}(\Gamma_0(240))

Newforms

LabelDimCoefficient fieldTrace formAL signsSignCMElliptic curve
240.4.a.a1Q\mathbb{Q} qโˆ’3q3โˆ’5q5โˆ’20q7+O(q8)q - 3q^{3} - 5q^{5} - 20q^{7} + O(q^{8})+ +
240.4.a.b1Q\mathbb{Q} qโˆ’3q3โˆ’5q5+4q7+O(q8)q - 3q^{3} - 5q^{5} + 4q^{7} + O(q^{8})+ +
240.4.a.c1Q\mathbb{Q} qโˆ’3q3+5q5โˆ’32q7+O(q8)q - 3q^{3} + 5q^{5} - 32q^{7} + O(q^{8})+ โˆ’
240.4.a.d1Q\mathbb{Q} qโˆ’3q3+5q5โˆ’8q7+O(q8)q - 3q^{3} + 5q^{5} - 8q^{7} + O(q^{8})+ โˆ’
240.4.a.e1Q\mathbb{Q} qโˆ’3q3+5q5+24q7+O(q8)q - 3q^{3} + 5q^{5} + 24q^{7} + O(q^{8})+ โˆ’
240.4.a.f1Q\mathbb{Q} q+3q3โˆ’5q5โˆ’20q7+O(q8)q + 3q^{3} - 5q^{5} - 20q^{7} + O(q^{8})โˆ’ +
240.4.a.g1Q\mathbb{Q} q+3q3โˆ’5q5โˆ’20q7+O(q8)q + 3q^{3} - 5q^{5} - 20q^{7} + O(q^{8})โˆ’ +
240.4.a.h1Q\mathbb{Q} q+3q3โˆ’5q5โˆ’4q7+O(q8)q + 3q^{3} - 5q^{5} - 4q^{7} + O(q^{8})โˆ’ +
240.4.a.i1Q\mathbb{Q} q+3q3โˆ’5q5+28q7+O(q8)q + 3q^{3} - 5q^{5} + 28q^{7} + O(q^{8})โˆ’ +
240.4.a.j1Q\mathbb{Q} q+3q3+5q5โˆ’32q7+O(q8)q + 3q^{3} + 5q^{5} - 32q^{7} + O(q^{8})โˆ’ โˆ’
240.4.a.k1Q\mathbb{Q} q+3q3+5q5+O(q8)q + 3q^{3} + 5q^{5} + O(q^{8})โˆ’ โˆ’
240.4.a.l1Q\mathbb{Q} q+3q3+5q5+16q7+O(q8)q + 3q^{3} + 5q^{5} + 16q^{7} + O(q^{8})โˆ’ โˆ’

Hecke characteristic polynomials

The eigenvalue of T=T7+3T11+5T13+7T17+9T19+11T23T = T_{7} + 3T_{11} + 5T_{13} + 7T_{17} + 9T_{19} + 11T_{23} 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
240.4.a.axโˆ’1332x - 1332
240.4.a.bx+2692x + 2692
240.4.a.cxโˆ’956x - 956
240.4.a.dx+1356x + 1356
240.4.a.exโˆ’1908x - 1908
240.4.a.fxโˆ’3236x - 3236
240.4.a.gx+2556x + 2556
240.4.a.hx+2124x + 2124
240.4.a.ixโˆ’1076x - 1076
240.4.a.jxโˆ’116x - 116
240.4.a.kx+396x + 396
240.4.a.lxโˆ’2436x - 2436

Dimensions

Newforms S4new(ฮ“0(240))S_{4}^{\mathrm{new}}(\Gamma_0(240))12
Old forms120
Cusp forms S4(ฮ“0(240))S_{4}(\Gamma_0(240))132
Eisenstein series E4(ฮ“0(240))E_{4}(\Gamma_0(240))24
Modular forms M4(ฮ“0(240))M_{4}(\Gamma_0(240))156

Decomposition of the old subspace

S4old(ฮ“0(240))=S_{4}^{\mathrm{old}}(\Gamma_0(240)) = S4new(ฮ“0(5))โŠ•10S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 10} โŠ•\oplus S4new(ฮ“0(6))โŠ•8S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 8} โŠ•\oplus S4new(ฮ“0(8))โŠ•8S_{4}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 8} โŠ•\oplus S4new(ฮ“0(10))โŠ•8S_{4}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 8} โŠ•\oplus S4new(ฮ“0(12))โŠ•6S_{4}^{\mathrm{new}}(\Gamma_0(12))^{\oplus 6} โŠ•\oplus S4new(ฮ“0(15))โŠ•5S_{4}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 5} โŠ•\oplus S4new(ฮ“0(16))โŠ•4S_{4}^{\mathrm{new}}(\Gamma_0(16))^{\oplus 4} โŠ•\oplus S4new(ฮ“0(20))โŠ•6S_{4}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 6} โŠ•\oplus S4new(ฮ“0(24))โŠ•4S_{4}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 4} โŠ•\oplus S4new(ฮ“0(30))โŠ•4S_{4}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 4} โŠ•\oplus S4new(ฮ“0(40))โŠ•4S_{4}^{\mathrm{new}}(\Gamma_0(40))^{\oplus 4} โŠ•\oplus S4new(ฮ“0(48))โŠ•2S_{4}^{\mathrm{new}}(\Gamma_0(48))^{\oplus 2} โŠ•\oplus S4new(ฮ“0(60))โŠ•3S_{4}^{\mathrm{new}}(\Gamma_0(60))^{\oplus 3} โŠ•\oplus S4new(ฮ“0(80))โŠ•2S_{4}^{\mathrm{new}}(\Gamma_0(80))^{\oplus 2} โŠ•\oplus S4new(ฮ“0(120))โŠ•2S_{4}^{\mathrm{new}}(\Gamma_0(120))^{\oplus 2}

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.67 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, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240]: [1, 2, 2, 3, 3, 5, 5, 7, 9, 8, 8, 12, 16, 22, 22, 28, 42, 40, 80, 152] (certified at ell = 2147483629)
  • T = [(7, 1), (11, 3), (13, 5), (17, 7), (19, 9), (23, 11)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 5 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(240, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 240:4 1000

โ† 239.4.a ยท 241.4.a โ†’