๐ŸŒฟ Sagebrush Atlas
Atlas โ€บ Modular forms โ€บ Weight 2 โ€บ Level 420 โ€บ 420.2.a

Newspace 420.2.a: S2new(ฮ“0(420))S_{2}^{\mathrm{new}}(\Gamma_0(420))

Newforms

LabelDimCoefficient fieldTrace formAL signsSignCMElliptic curve
420.2.a.a1Q\mathbb{Q} qโˆ’q3โˆ’q5โˆ’q7+O(q8)q - q^{3} - q^{5} - q^{7} + O(q^{8})+ + + 420a1 rank 0
420.2.a.b1Q\mathbb{Q} qโˆ’q3+q5+q7+O(q8)q - q^{3} + q^{5} + q^{7} + O(q^{8})+ โˆ’ โˆ’ 420b1 rank 0
420.2.a.c1Q\mathbb{Q} q+q3โˆ’q5+q7+O(q8)q + q^{3} - q^{5} + q^{7} + O(q^{8})โˆ’ + โˆ’ 420c1 rank 0
420.2.a.d1Q\mathbb{Q} q+q3+q5โˆ’q7+O(q8)q + q^{3} + q^{5} - q^{7} + O(q^{8})โˆ’ โˆ’ + 420d1 rank 0

Hecke characteristic polynomials

The eigenvalue of T=T11+3T13+5T17+7T19+9T23+11T29T = T_{11} + 3T_{13} + 5T_{17} + 7T_{19} + 9T_{23} + 11T_{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
420.2.a.ax+8x + 8
420.2.a.bxโˆ’136x - 136
420.2.a.cxโˆ’104x - 104
420.2.a.dxโˆ’24x - 24

Dimensions

Newforms S2new(ฮ“0(420))S_{2}^{\mathrm{new}}(\Gamma_0(420))4
Old forms81
Cusp forms S2(ฮ“0(420))S_{2}(\Gamma_0(420))85
Eisenstein series E2(ฮ“0(420))E_{2}(\Gamma_0(420))23
Modular forms M2(ฮ“0(420))M_{2}(\Gamma_0(420))108

Decomposition of the old subspace

S2old(ฮ“0(420))=S_{2}^{\mathrm{old}}(\Gamma_0(420)) = S2new(ฮ“0(14))โŠ•8S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 8} โŠ•\oplus 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(21))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 6} โŠ•\oplus S2new(ฮ“0(30))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(35))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 6} โŠ•\oplus S2new(ฮ“0(42))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(42))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(70))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(70))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(84))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(84))^{\oplus 2} โŠ•\oplus S2new(ฮ“0(105))โŠ•3S_{2}^{\mathrm{new}}(\Gamma_0(105))^{\oplus 3} โŠ•\oplus S2new(ฮ“0(140))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(140))^{\oplus 2} โŠ•\oplus S2new(ฮ“0(210))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(210))^{\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, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420]: [0, 1, 1, 2, 1, 3, 1, 3, 5, 4, 4, 6, 4, 7, 10, 6, 12, 18, 16, 22, 20, 30, 56, 108] (certified at ell = 2147483629)
  • T = [(11, 1), (13, 3), (17, 5), (19, 7), (23, 9), (29, 11)]; h of degree 4 = 1 x 4 (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(420, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 420:2 1000

โ† 419.2.a ยท 421.2.a โ†’