๐ŸŒฟ Sagebrush Atlas
Atlas โ€บ Modular forms โ€บ Weight 2 โ€บ Level 900 โ€บ 900.2.a

Newspace 900.2.a: S2new(ฮ“0(900))S_{2}^{\mathrm{new}}(\Gamma_0(900))

Newforms

LabelDimCoefficient fieldTrace formAL signsSignCMElliptic curve
900.2.a.a1Q\mathbb{Q} qโˆ’4q7+O(q8)q - 4q^{7} + O(q^{8}) 900h1 rank 0
900.2.a.b1Q\mathbb{Q} qโˆ’2q7+O(q8)q - 2q^{7} + O(q^{8}) 900e1 rank 1
900.2.a.c1Q\mathbb{Q} qโˆ’q7+O(q8)q - q^{7} + O(q^{8}) 900f1 rank 0
900.2.a.d1Q\mathbb{Q} qโˆ’q7+O(q8)q - q^{7} + O(q^{8}) Q(โˆ’3)\mathbb{Q}(\sqrt{-3})900c1 rank 1
900.2.a.e1Q\mathbb{Q} q+q7+O(q8)q + q^{7} + O(q^{8}) 900d1 rank 1
900.2.a.f1Q\mathbb{Q} q+q7+O(q8)q + q^{7} + O(q^{8}) Q(โˆ’3)\mathbb{Q}(\sqrt{-3})900a1 rank 0
900.2.a.g1Q\mathbb{Q} q+4q7+O(q8)q + 4q^{7} + O(q^{8}) Q(โˆ’3)\mathbb{Q}(\sqrt{-3})900b1 rank 0
900.2.a.h1Q\mathbb{Q} q+4q7+O(q8)q + 4q^{7} + O(q^{8}) 900g1 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
900.2.a.axโˆ’118x - 118
900.2.a.bx+90x + 90
900.2.a.cxโˆ’121x - 121
900.2.a.dxโˆ’31x - 31
900.2.a.ex+27x + 27
900.2.a.fxโˆ’75x - 75
900.2.a.gxโˆ’6x - 6
900.2.a.hxโˆ’46x - 46

Dimensions

Newforms S2new(ฮ“0(900))S_{2}^{\mathrm{new}}(\Gamma_0(900))8
Old forms137
Cusp forms S2(ฮ“0(900))S_{2}(\Gamma_0(900))145
Eisenstein series E2(ฮ“0(900))E_{2}(\Gamma_0(900))71
Modular forms M2(ฮ“0(900))M_{2}(\Gamma_0(900))216

Decomposition of the old subspace

S2old(ฮ“0(900))=S_{2}^{\mathrm{old}}(\Gamma_0(900)) = S2new(ฮ“0(15))โŠ•12S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 12} โŠ•\oplus S2new(ฮ“0(20))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 6} โŠ•\oplus S2new(ฮ“0(30))โŠ•8S_{2}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 8} โŠ•\oplus S2new(ฮ“0(36))โŠ•3S_{2}^{\mathrm{new}}(\Gamma_0(36))^{\oplus 3} โŠ•\oplus S2new(ฮ“0(45))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(45))^{\oplus 6} โŠ•\oplus S2new(ฮ“0(50))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(50))^{\oplus 6} โŠ•\oplus S2new(ฮ“0(75))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(75))^{\oplus 6} โŠ•\oplus S2new(ฮ“0(90))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(90))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(100))โŠ•3S_{2}^{\mathrm{new}}(\Gamma_0(100))^{\oplus 3} โŠ•\oplus S2new(ฮ“0(150))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(150))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(180))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(180))^{\oplus 2} โŠ•\oplus S2new(ฮ“0(225))โŠ•3S_{2}^{\mathrm{new}}(\Gamma_0(225))^{\oplus 3} โŠ•\oplus S2new(ฮ“0(300))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(300))^{\oplus 2} โŠ•\oplus S2new(ฮ“0(450))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(450))^{\oplus 2}

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.959 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, 25, 30, 36, 45, 50, 60, 75, 90, 100, 150, 180, 225, 300, 450, 900]: [0, 1, 1, 2, 1, 3, 2, 3, 5, 4, 5, 6, 3, 10, 9, 8, 9, 18, 12, 22, 18, 34, 42, 32, 66, 94, 186] (certified at ell = 2147482921)
  • T = [(7, 1), (11, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8)]; 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(900, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 900:2 1000

โ† 899.2.a ยท 901.2.a โ†’