๐ŸŒฟ Sagebrush Atlas
Atlas โ€บ Modular forms โ€บ Weight 4 โ€บ Level 144 โ€บ 144.4.a

Newspace 144.4.a: S4new(ฮ“0(144))S_{4}^{\mathrm{new}}(\Gamma_0(144))

Newforms

LabelDimCoefficient fieldTrace formAL signsSignCMElliptic curve
144.4.a.a1Q\mathbb{Q} qโˆ’16q5+12q7+O(q8)q - 16q^{5} + 12q^{7} + O(q^{8})
144.4.a.b1Q\mathbb{Q} qโˆ’14q5+24q7+O(q8)q - 14q^{5} + 24q^{7} + O(q^{8})
144.4.a.c1Q\mathbb{Q} qโˆ’6q5+16q7+O(q8)q - 6q^{5} + 16q^{7} + O(q^{8})
144.4.a.d1Q\mathbb{Q} qโˆ’20q7+O(q8)q - 20q^{7} + O(q^{8}) Q(โˆ’3)\mathbb{Q}(\sqrt{-3})
144.4.a.e1Q\mathbb{Q} q+2q5โˆ’24q7+O(q8)q + 2q^{5} - 24q^{7} + O(q^{8})
144.4.a.f1Q\mathbb{Q} q+16q5+12q7+O(q8)q + 16q^{5} + 12q^{7} + O(q^{8})
144.4.a.g1Q\mathbb{Q} q+18q5โˆ’8q7+O(q8)q + 18q^{5} - 8q^{7} + O(q^{8})

Hecke characteristic polynomials

The eigenvalue of T=T5+2T7+3T13+4T17+5T19+6T11+7T23+8T29T = T_{5} + 2T_{7} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{11} + 7T_{23} + 8T_{29} 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
144.4.a.axโˆ’1374x - 1374
144.4.a.bxโˆ’16x - 16
144.4.a.cxโˆ’1552x - 1552
144.4.a.dx+530x + 530
144.4.a.ex+2640x + 2640
144.4.a.fxโˆ’382x - 382
144.4.a.gxโˆ’2992x - 2992

Dimensions

Newforms S4new(ฮ“0(144))S_{4}^{\mathrm{new}}(\Gamma_0(144))7
Old forms53
Cusp forms S4(ฮ“0(144))S_{4}(\Gamma_0(144))60
Eisenstein series E4(ฮ“0(144))E_{4}(\Gamma_0(144))24
Modular forms M4(ฮ“0(144))M_{4}(\Gamma_0(144))84

Decomposition of the old subspace

S4old(ฮ“0(144))=S_{4}^{\mathrm{old}}(\Gamma_0(144)) = S4new(ฮ“0(6))โŠ•8S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 8} โŠ•\oplus S4new(ฮ“0(8))โŠ•6S_{4}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 6} โŠ•\oplus S4new(ฮ“0(9))โŠ•5S_{4}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 5} โŠ•\oplus S4new(ฮ“0(12))โŠ•6S_{4}^{\mathrm{new}}(\Gamma_0(12))^{\oplus 6} โŠ•\oplus S4new(ฮ“0(16))โŠ•3S_{4}^{\mathrm{new}}(\Gamma_0(16))^{\oplus 3} โŠ•\oplus S4new(ฮ“0(18))โŠ•4S_{4}^{\mathrm{new}}(\Gamma_0(18))^{\oplus 4} โŠ•\oplus S4new(ฮ“0(24))โŠ•4S_{4}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 4} โŠ•\oplus S4new(ฮ“0(36))โŠ•3S_{4}^{\mathrm{new}}(\Gamma_0(36))^{\oplus 3} โŠ•\oplus S4new(ฮ“0(48))โŠ•2S_{4}^{\mathrm{new}}(\Gamma_0(48))^{\oplus 2} โŠ•\oplus S4new(ฮ“0(72))โŠ•2S_{4}^{\mathrm{new}}(\Gamma_0(72))^{\oplus 2}

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.303 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, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144]: [1, 2, 2, 3, 5, 5, 4, 9, 8, 11, 16, 21, 28, 40, 76] (certified at ell = 2147483629)
  • T = [(5, 1), (7, 2), (13, 3), (17, 4), (19, 5), (11, 6), (23, 7), (29, 8)]; h of degree 7 = 1 x 7 (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(144, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 144:4 1000

โ† 143.4.a ยท 145.4.a โ†’