๐ŸŒฟ Sagebrush Atlas
Atlas โ€บ Modular forms โ€บ Weight 2 โ€บ Level 756 โ€บ 756.2.a

Newspace 756.2.a: S2new(ฮ“0(756))S_{2}^{\mathrm{new}}(\Gamma_0(756))

Newforms

LabelDimCoefficient fieldTrace formAL signsSignCMElliptic curve
756.2.a.a1Q\mathbb{Q} qโˆ’3q5+q7+O(q8)q - 3q^{5} + q^{7} + O(q^{8})โˆ’ 756f1 rank 0
756.2.a.b1Q\mathbb{Q} qโˆ’3q5+q7+O(q8)q - 3q^{5} + q^{7} + O(q^{8})โˆ’ 756b1 rank 1
756.2.a.c1Q\mathbb{Q} qโˆ’q5โˆ’q7+O(q8)q - q^{5} - q^{7} + O(q^{8})+ 756c1 rank 1
756.2.a.d1Q\mathbb{Q} q+q5โˆ’q7+O(q8)q + q^{5} - q^{7} + O(q^{8})+ 756a1 rank 0
756.2.a.e1Q\mathbb{Q} q+3q5+q7+O(q8)q + 3q^{5} + q^{7} + O(q^{8})โˆ’ 756d1 rank 0
756.2.a.f1Q\mathbb{Q} q+3q5+q7+O(q8)q + 3q^{5} + q^{7} + O(q^{8})โˆ’ 756e1 rank 0
756.2.a.g2Q(13)\mathbb{Q}(\sqrt{13}) 2qโˆ’2q7+O(q8)2q - 2q^{7} + O(q^{8})+

Hecke characteristic polynomials

The eigenvalue of T=T5+3T11T = T_{5} + 3T_{11} 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
756.2.a.ax+12x + 12
756.2.a.bx+3x + 3
756.2.a.cx+7x + 7
756.2.a.dxโˆ’7x - 7
756.2.a.exโˆ’3x - 3
756.2.a.fxโˆ’12x - 12
756.2.a.gx2โˆ’52x^{2} - 52

Dimensions

Newforms S2new(ฮ“0(756))S_{2}^{\mathrm{new}}(\Gamma_0(756))8
Old forms119
Cusp forms S2(ฮ“0(756))S_{2}(\Gamma_0(756))127
Eisenstein series E2(ฮ“0(756))E_{2}(\Gamma_0(756))35
Modular forms M2(ฮ“0(756))M_{2}(\Gamma_0(756))162

Decomposition of the old subspace

S2old(ฮ“0(756))=S_{2}^{\mathrm{old}}(\Gamma_0(756)) = S2new(ฮ“0(14))โŠ•8S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 8} โŠ•\oplus S2new(ฮ“0(21))โŠ•9S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 9} โŠ•\oplus S2new(ฮ“0(27))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 6} 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(54))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(54))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(63))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(63))^{\oplus 6} โŠ•\oplus S2new(ฮ“0(84))โŠ•3S_{2}^{\mathrm{new}}(\Gamma_0(84))^{\oplus 3} โŠ•\oplus S2new(ฮ“0(108))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(108))^{\oplus 2} โŠ•\oplus S2new(ฮ“0(126))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(126))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(189))โŠ•3S_{2}^{\mathrm{new}}(\Gamma_0(189))^{\oplus 3} โŠ•\oplus S2new(ฮ“0(252))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(252))^{\oplus 2} โŠ•\oplus S2new(ฮ“0(378))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(378))^{\oplus 2}

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.507 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, 9, 12, 14, 18, 21, 27, 28, 36, 42, 54, 63, 84, 108, 126, 189, 252, 378, 756]: [0, 1, 1, 2, 3, 1, 2, 5, 4, 5, 4, 4, 7, 9, 12, 11, 10, 22, 21, 28, 26, 54, 76, 150] (certified at ell = 2147483647)
  • T = [(5, 1), (11, 3)]; h of degree 8 = 1 x 8 (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(756, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 756:2 1000

โ† 755.2.a ยท 757.2.a โ†’