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Atlas โ€บ Modular forms โ€บ Weight 4 โ€บ Level 180 โ€บ 180.4.a

Newspace 180.4.a: S4new(ฮ“0(180))S_{4}^{\mathrm{new}}(\Gamma_0(180))

Newforms

LabelDimCoefficient fieldTrace formAL signsSignCMElliptic curve
180.4.a.a1Q\mathbb{Q} qโˆ’5q5โˆ’16q7+O(q8)q - 5q^{5} - 16q^{7} + O(q^{8})+
180.4.a.b1Q\mathbb{Q} qโˆ’5q5+2q7+O(q8)q - 5q^{5} + 2q^{7} + O(q^{8})+
180.4.a.c1Q\mathbb{Q} qโˆ’5q5+32q7+O(q8)q - 5q^{5} + 32q^{7} + O(q^{8})+
180.4.a.d1Q\mathbb{Q} q+5q5โˆ’28q7+O(q8)q + 5q^{5} - 28q^{7} + O(q^{8})โˆ’
180.4.a.e1Q\mathbb{Q} q+5q5+2q7+O(q8)q + 5q^{5} + 2q^{7} + O(q^{8})โˆ’

Hecke characteristic polynomials

The eigenvalue of T=T7+3T11+5T13+7T17+9T19T = T_{7} + 3T_{11} + 5T_{13} + 7T_{17} + 9T_{19} 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
180.4.a.axโˆ’864x - 864
180.4.a.bx+990x + 990
180.4.a.cxโˆ’1680x - 1680
180.4.a.dx+840x + 840
180.4.a.exโˆ’450x - 450

Dimensions

Newforms S4new(ฮ“0(180))S_{4}^{\mathrm{new}}(\Gamma_0(180))5
Old forms91
Cusp forms S4(ฮ“0(180))S_{4}(\Gamma_0(180))96
Eisenstein series E4(ฮ“0(180))E_{4}(\Gamma_0(180))24
Modular forms M4(ฮ“0(180))M_{4}(\Gamma_0(180))120

Decomposition of the old subspace

S4old(ฮ“0(180))=S_{4}^{\mathrm{old}}(\Gamma_0(180)) = S4new(ฮ“0(5))โŠ•9S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 9} โŠ•\oplus S4new(ฮ“0(6))โŠ•8S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 8} โŠ•\oplus S4new(ฮ“0(9))โŠ•6S_{4}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 6} โŠ•\oplus S4new(ฮ“0(10))โŠ•6S_{4}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 6} โŠ•\oplus S4new(ฮ“0(12))โŠ•4S_{4}^{\mathrm{new}}(\Gamma_0(12))^{\oplus 4} โŠ•\oplus S4new(ฮ“0(15))โŠ•6S_{4}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 6} โŠ•\oplus S4new(ฮ“0(18))โŠ•4S_{4}^{\mathrm{new}}(\Gamma_0(18))^{\oplus 4} โŠ•\oplus S4new(ฮ“0(20))โŠ•3S_{4}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 3} โŠ•\oplus S4new(ฮ“0(30))โŠ•4S_{4}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 4} โŠ•\oplus S4new(ฮ“0(36))โŠ•2S_{4}^{\mathrm{new}}(\Gamma_0(36))^{\oplus 2} โŠ•\oplus S4new(ฮ“0(45))โŠ•3S_{4}^{\mathrm{new}}(\Gamma_0(45))^{\oplus 3} โŠ•\oplus S4new(ฮ“0(60))โŠ•2S_{4}^{\mathrm{new}}(\Gamma_0(60))^{\oplus 2} โŠ•\oplus S4new(ฮ“0(90))โŠ•2S_{4}^{\mathrm{new}}(\Gamma_0(90))^{\oplus 2}

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.279 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, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180]: [1, 2, 2, 3, 3, 5, 4, 7, 9, 8, 11, 12, 22, 21, 20, 42, 58, 114] (certified at ell = 2147483629)
  • T = [(7, 1), (11, 3), (13, 5), (17, 7), (19, 9)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 2 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(180, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 180:4 1000

โ† 179.4.a ยท 181.4.a โ†’