๐ŸŒฟ Sagebrush Atlas
Atlas โ€บ Modular forms โ€บ Weight 2 โ€บ Level 504 โ€บ 504.2.a

Newspace 504.2.a: S2new(ฮ“0(504))S_{2}^{\mathrm{new}}(\Gamma_0(504))

Newforms

LabelDimCoefficient fieldTrace formAL signsSignCMElliptic curve
504.2.a.a1Q\mathbb{Q} qโˆ’2q5โˆ’q7+O(q8)q - 2q^{5} - q^{7} + O(q^{8})+ 504a1 rank 1
504.2.a.b1Q\mathbb{Q} qโˆ’2q5โˆ’q7+O(q8)q - 2q^{5} - q^{7} + O(q^{8})+ 504f1 rank 1
504.2.a.c1Q\mathbb{Q} qโˆ’2q5โˆ’q7+O(q8)q - 2q^{5} - q^{7} + O(q^{8})+ 504c1 rank 0
504.2.a.d1Q\mathbb{Q} qโˆ’2q5+q7+O(q8)q - 2q^{5} + q^{7} + O(q^{8})โˆ’ 504e1 rank 1
504.2.a.e1Q\mathbb{Q} qโˆ’2q5+q7+O(q8)q - 2q^{5} + q^{7} + O(q^{8})โˆ’ 504g1 rank 0
504.2.a.f1Q\mathbb{Q} q+2q5โˆ’q7+O(q8)q + 2q^{5} - q^{7} + O(q^{8})+ 504d1 rank 0
504.2.a.g1Q\mathbb{Q} q+2q5+q7+O(q8)q + 2q^{5} + q^{7} + O(q^{8})โˆ’ 504b1 rank 0
504.2.a.h1Q\mathbb{Q} q+4q5+q7+O(q8)q + 4q^{5} + q^{7} + O(q^{8})โˆ’ 504h1 rank 0

Hecke characteristic polynomials

The eigenvalue of T=T5+2T11+3T13+4T17+5T19+6T23+7T29+8T31+9T37+10T41+11T43+12T47T = T_{5} + 2T_{11} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47} 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
504.2.a.ax+134x + 134
504.2.a.bxโˆ’104x - 104
504.2.a.cxโˆ’112x - 112
504.2.a.dx+190x + 190
504.2.a.exโˆ’8x - 8
504.2.a.fxโˆ’134x - 134
504.2.a.gxโˆ’62x - 62
504.2.a.hxโˆ’74x - 74

Dimensions

Newforms S2new(ฮ“0(504))S_{2}^{\mathrm{new}}(\Gamma_0(504))8
Old forms73
Cusp forms S2(ฮ“0(504))S_{2}(\Gamma_0(504))81
Eisenstein series E2(ฮ“0(504))E_{2}(\Gamma_0(504))31
Modular forms M2(ฮ“0(504))M_{2}(\Gamma_0(504))112

Decomposition of the old subspace

S2old(ฮ“0(504))=S_{2}^{\mathrm{old}}(\Gamma_0(504)) = S2new(ฮ“0(14))โŠ•9S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 9} โŠ•\oplus S2new(ฮ“0(21))โŠ•8S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 8} โŠ•\oplus S2new(ฮ“0(24))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(36))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(36))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(42))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(42))^{\oplus 6} โŠ•\oplus S2new(ฮ“0(56))โŠ•3S_{2}^{\mathrm{new}}(\Gamma_0(56))^{\oplus 3} โŠ•\oplus S2new(ฮ“0(63))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(63))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(72))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(72))^{\oplus 2} โŠ•\oplus S2new(ฮ“0(84))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(84))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(126))โŠ•3S_{2}^{\mathrm{new}}(\Gamma_0(126))^{\oplus 3} โŠ•\oplus S2new(ฮ“0(168))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(168))^{\oplus 2} โŠ•\oplus S2new(ฮ“0(252))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(252))^{\oplus 2}

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.005 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, 6, 7, 8, 9, 12, 14, 18, 21, 24, 28, 36, 42, 56, 63, 72, 84, 126, 168, 252, 504]: [0, 1, 1, 2, 3, 1, 3, 2, 5, 4, 5, 4, 8, 7, 9, 12, 12, 10, 16, 22, 28, 40, 54, 104] (certified at ell = 2147483647)
  • T = [(5, 1), (11, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; 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(504, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 504:2 1000

โ† 503.2.a ยท 505.2.a โ†’