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Atlas โ€บ Modular forms โ€บ Weight 2 โ€บ Level 360 โ€บ 360.2.a

Newspace 360.2.a: S2new(ฮ“0(360))S_{2}^{\mathrm{new}}(\Gamma_0(360))

Newforms

LabelDimCoefficient fieldTrace formAL signsSignCMElliptic curve
360.2.a.a1Q\mathbb{Q} qโˆ’q5โˆ’4q7+O(q8)q - q^{5} - 4q^{7} + O(q^{8})+ 360e1 rank 1
360.2.a.b1Q\mathbb{Q} qโˆ’q5+O(q8)q - q^{5} + O(q^{8})+ 360a1 rank 0
360.2.a.c1Q\mathbb{Q} qโˆ’q5+2q7+O(q8)q - q^{5} + 2q^{7} + O(q^{8})+ 360b1 rank 0
360.2.a.d1Q\mathbb{Q} q+q5+2q7+O(q8)q + q^{5} + 2q^{7} + O(q^{8})โˆ’ 360c1 rank 0
360.2.a.e1Q\mathbb{Q} q+q5+4q7+O(q8)q + q^{5} + 4q^{7} + O(q^{8})โˆ’ 360d1 rank 0

Hecke characteristic polynomials

The eigenvalue of T=T7+2T11+3T13+4T17+5T19+6T23+7T29+8T31T = T_{7} + 2T_{11} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{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
360.2.a.ax+80x + 80
360.2.a.bx+20x + 20
360.2.a.cxโˆ’40x - 40
360.2.a.dxโˆ’92x - 92
360.2.a.exโˆ’104x - 104

Dimensions

Newforms S2new(ฮ“0(360))S_{2}^{\mathrm{new}}(\Gamma_0(360))5
Old forms52
Cusp forms S2(ฮ“0(360))S_{2}(\Gamma_0(360))57
Eisenstein series E2(ฮ“0(360))E_{2}(\Gamma_0(360))31
Modular forms M2(ฮ“0(360))M_{2}(\Gamma_0(360))88

Decomposition of the old subspace

S2old(ฮ“0(360))=S_{2}^{\mathrm{old}}(\Gamma_0(360)) = S2new(ฮ“0(15))โŠ•8S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 8} โŠ•\oplus S2new(ฮ“0(20))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 6} โŠ•\oplus S2new(ฮ“0(24))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 4} 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(40))โŠ•3S_{2}^{\mathrm{new}}(\Gamma_0(40))^{\oplus 3} โŠ•\oplus S2new(ฮ“0(45))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(45))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(72))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(72))^{\oplus 2} โŠ•\oplus S2new(ฮ“0(90))โŠ•3S_{2}^{\mathrm{new}}(\Gamma_0(90))^{\oplus 3} โŠ•\oplus S2new(ฮ“0(120))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(120))^{\oplus 2} โŠ•\oplus S2new(ฮ“0(180))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(180))^{\oplus 2}

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.427 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, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360]: [0, 1, 1, 2, 1, 3, 3, 2, 3, 5, 4, 5, 6, 8, 10, 9, 10, 8, 18, 16, 22, 32, 42, 80] (certified at ell = 2147483629)
  • T = [(7, 1), (11, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8)]; 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(360, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 360:2 1000

โ† 359.2.a ยท 361.2.a โ†’