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Atlas โ€บ Modular forms โ€บ Weight 2 โ€บ Level 660 โ€บ 660.2.a

Newspace 660.2.a: S2new(ฮ“0(660))S_{2}^{\mathrm{new}}(\Gamma_0(660))

Newforms

LabelDimCoefficient fieldTrace formAL signsSignCMElliptic curve
660.2.a.a1Q\mathbb{Q} qโˆ’q3โˆ’q5โˆ’2q7+O(q8)q - q^{3} - q^{5} - 2q^{7} + O(q^{8})+ + + 660a1 rank 0
660.2.a.b1Q\mathbb{Q} qโˆ’q3โˆ’q5+O(q8)q - q^{3} - q^{5} + O(q^{8})+ + โˆ’ 660b1 rank 1
660.2.a.c1Q\mathbb{Q} q+q3โˆ’q5โˆ’4q7+O(q8)q + q^{3} - q^{5} - 4q^{7} + O(q^{8})โˆ’ + + 660c1 rank 1
660.2.a.d1Q\mathbb{Q} q+q3โˆ’q5+2q7+O(q8)q + q^{3} - q^{5} + 2q^{7} + O(q^{8})โˆ’ + โˆ’ 660d1 rank 0
660.2.a.e2Q(13)\mathbb{Q}(\sqrt{13}) 2qโˆ’2q3+2q5+2q7+O(q8)2q - 2q^{3} + 2q^{5} + 2q^{7} + O(q^{8})+ โˆ’ โˆ’
660.2.a.f2Q(13)\mathbb{Q}(\sqrt{13}) 2q+2q3+2q5+2q7+O(q8)2q + 2q^{3} + 2q^{5} + 2q^{7} + O(q^{8})โˆ’ โˆ’ +

Hecke characteristic polynomials

The eigenvalue of T=T7+3T13+5T17+7T19+9T23+11T29T = T_{7} + 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
660.2.a.axโˆ’102x - 102
660.2.a.bx+124x + 124
660.2.a.cx+32x + 32
660.2.a.dxโˆ’22x - 22
660.2.a.ex2โˆ’202x+8628x^{2} - 202x + 8628
660.2.a.fx2โˆ’102xโˆ’8332x^{2} - 102x - 8332

Dimensions

Newforms S2new(ฮ“0(660))S_{2}^{\mathrm{new}}(\Gamma_0(660))8
Old forms125
Cusp forms S2(ฮ“0(660))S_{2}(\Gamma_0(660))133
Eisenstein series E2(ฮ“0(660))E_{2}(\Gamma_0(660))23
Modular forms M2(ฮ“0(660))M_{2}(\Gamma_0(660))156

Decomposition of the old subspace

S2old(ฮ“0(660))=S_{2}^{\mathrm{old}}(\Gamma_0(660)) = S2new(ฮ“0(11))โŠ•12S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 12} 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(30))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(33))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 6} โŠ•\oplus S2new(ฮ“0(44))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(44))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(55))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(55))^{\oplus 6} โŠ•\oplus S2new(ฮ“0(66))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(66))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(110))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(110))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(132))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(132))^{\oplus 2} โŠ•\oplus S2new(ฮ“0(165))โŠ•3S_{2}^{\mathrm{new}}(\Gamma_0(165))^{\oplus 3} โŠ•\oplus S2new(ฮ“0(220))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(220))^{\oplus 2} โŠ•\oplus S2new(ฮ“0(330))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(330))^{\oplus 2}

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.631 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, 10, 11, 12, 15, 20, 22, 30, 33, 44, 55, 60, 66, 110, 132, 165, 220, 330, 660]: [0, 1, 1, 2, 1, 3, 3, 2, 5, 4, 6, 5, 10, 6, 9, 8, 18, 16, 22, 30, 28, 42, 80, 156] (certified at ell = 2147482921)
  • T = [(7, 1), (13, 3), (17, 5), (19, 7), (23, 9), (29, 11)]; h of degree 8 = 1 x 8 (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(660, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 660:2 1000

โ† 659.2.a ยท 661.2.a โ†’