๐ŸŒฟ Sagebrush Atlas
Atlas โ€บ Modular forms โ€บ Weight 2 โ€บ Level 540 โ€บ 540.2.a

Newspace 540.2.a: S2new(ฮ“0(540))S_{2}^{\mathrm{new}}(\Gamma_0(540))

Newforms

LabelDimCoefficient fieldTrace formAL signsSignCMElliptic curve
540.2.a.a1Q\mathbb{Q} qโˆ’q5โˆ’4q7+O(q8)q - q^{5} - 4q^{7} + O(q^{8})+ 540d1 rank 1
540.2.a.b1Q\mathbb{Q} qโˆ’q5โˆ’q7+O(q8)q - q^{5} - q^{7} + O(q^{8})+ 540c1 rank 1
540.2.a.c1Q\mathbb{Q} qโˆ’q5+2q7+O(q8)q - q^{5} + 2q^{7} + O(q^{8})+ 540a1 rank 0
540.2.a.d1Q\mathbb{Q} q+q5โˆ’4q7+O(q8)q + q^{5} - 4q^{7} + O(q^{8})โˆ’ 540b1 rank 1
540.2.a.e1Q\mathbb{Q} q+q5โˆ’q7+O(q8)q + q^{5} - q^{7} + O(q^{8})โˆ’ 540e1 rank 0
540.2.a.f1Q\mathbb{Q} q+q5+2q7+O(q8)q + q^{5} + 2q^{7} + O(q^{8})โˆ’ 540f1 rank 0

Hecke characteristic polynomials

The eigenvalue of T=6146T7+2849T11+7198T13+8490T17+134T19+8011T23+7196T29+3588T31T = 6146T_{7} + 2849T_{11} + 7198T_{13} + 8490T_{17} + 134T_{19} + 8011T_{23} + 7196T_{29} + 3588T_{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
540.2.a.ax+159905x + 159905
540.2.a.bx+93110x + 93110
540.2.a.cxโˆ’89911x - 89911
540.2.a.dxโˆ’1045x - 1045
540.2.a.exโˆ’123562x - 123562
540.2.a.fxโˆ’685x - 685

Dimensions

Newforms S2new(ฮ“0(540))S_{2}^{\mathrm{new}}(\Gamma_0(540))6
Old forms85
Cusp forms S2(ฮ“0(540))S_{2}(\Gamma_0(540))91
Eisenstein series E2(ฮ“0(540))E_{2}(\Gamma_0(540))35
Modular forms M2(ฮ“0(540))M_{2}(\Gamma_0(540))126

Decomposition of the old subspace

S2old(ฮ“0(540))=S_{2}^{\mathrm{old}}(\Gamma_0(540)) = S2new(ฮ“0(15))โŠ•9S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 9} โŠ•\oplus S2new(ฮ“0(20))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(27))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 6} โŠ•\oplus S2new(ฮ“0(30))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 6} โŠ•\oplus S2new(ฮ“0(36))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(36))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(45))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(45))^{\oplus 6} โŠ•\oplus S2new(ฮ“0(54))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(54))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(90))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(90))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(108))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(108))^{\oplus 2} โŠ•\oplus S2new(ฮ“0(135))โŠ•3S_{2}^{\mathrm{new}}(\Gamma_0(135))^{\oplus 3} โŠ•\oplus S2new(ฮ“0(180))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(180))^{\oplus 2} โŠ•\oplus S2new(ฮ“0(270))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(270))^{\oplus 2}

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.21 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, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, 270, 540]: [0, 1, 1, 2, 1, 3, 2, 3, 5, 4, 5, 6, 4, 10, 9, 8, 11, 18, 22, 21, 20, 42, 58, 114] (certified at ell = 2147483629)
  • T = [(7, 6146), (11, 2849), (13, 7198), (17, 8490), (19, 134), (23, 8011), (29, 7196), (31, 3588)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 4 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(540, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 540:2 1000

โ† 539.2.a ยท 541.2.a โ†’