๐ŸŒฟ Sagebrush Atlas
Atlas โ€บ Modular forms โ€บ Weight 2 โ€บ Level 480 โ€บ 480.2.a

Newspace 480.2.a: S2new(ฮ“0(480))S_{2}^{\mathrm{new}}(\Gamma_0(480))

Newforms

LabelDimCoefficient fieldTrace formAL signsSignCMElliptic curve
480.2.a.a1Q\mathbb{Q} qโˆ’q3โˆ’q5โˆ’4q7+O(q8)q - q^{3} - q^{5} - 4q^{7} + O(q^{8})+ + 480e1 rank 0
480.2.a.b1Q\mathbb{Q} qโˆ’q3โˆ’q5+O(q8)q - q^{3} - q^{5} + O(q^{8})+ + 480a1 rank 1
480.2.a.c1Q\mathbb{Q} qโˆ’q3+q5โˆ’4q7+O(q8)q - q^{3} + q^{5} - 4q^{7} + O(q^{8})+ โˆ’ 480f1 rank 1
480.2.a.d1Q\mathbb{Q} qโˆ’q3+q5+O(q8)q - q^{3} + q^{5} + O(q^{8})+ โˆ’ 480b1 rank 0
480.2.a.e1Q\mathbb{Q} q+q3โˆ’q5+O(q8)q + q^{3} - q^{5} + O(q^{8})โˆ’ + 480c1 rank 0
480.2.a.f1Q\mathbb{Q} q+q3โˆ’q5+4q7+O(q8)q + q^{3} - q^{5} + 4q^{7} + O(q^{8})โˆ’ + 480d1 rank 0
480.2.a.g1Q\mathbb{Q} q+q3+q5+O(q8)q + q^{3} + q^{5} + O(q^{8})โˆ’ โˆ’ 480g1 rank 0
480.2.a.h1Q\mathbb{Q} q+q3+q5+4q7+O(q8)q + q^{3} + q^{5} + 4q^{7} + O(q^{8})โˆ’ โˆ’ 480h1 rank 0

Hecke characteristic polynomials

The eigenvalue of T=T7+2T11+3T13+4T17+5T19+6T23+7T29+8T31+9T37+10T41+11T43+12T47+13T53+14T59+15T61+16T67T = T_{7} + 2T_{11} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47} + 13T_{53} + 14T_{59} + 15T_{61} + 16T_{67} 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
480.2.a.axโˆ’306x - 306
480.2.a.bxโˆ’98x - 98
480.2.a.cx+514x + 514
480.2.a.dxโˆ’102x - 102
480.2.a.ex+6x + 6
480.2.a.fx+502x + 502
480.2.a.gxโˆ’38x - 38
480.2.a.hx+146x + 146

Dimensions

Newforms S2new(ฮ“0(480))S_{2}^{\mathrm{new}}(\Gamma_0(480))8
Old forms73
Cusp forms S2(ฮ“0(480))S_{2}(\Gamma_0(480))81
Eisenstein series E2(ฮ“0(480))E_{2}(\Gamma_0(480))31
Modular forms M2(ฮ“0(480))M_{2}(\Gamma_0(480))112

Decomposition of the old subspace

S2old(ฮ“0(480))=S_{2}^{\mathrm{old}}(\Gamma_0(480)) = S2new(ฮ“0(15))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 6} โŠ•\oplus S2new(ฮ“0(20))โŠ•8S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 8} โŠ•\oplus S2new(ฮ“0(24))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 6} โŠ•\oplus S2new(ฮ“0(30))โŠ•5S_{2}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 5} โŠ•\oplus S2new(ฮ“0(32))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(40))โŠ•6S_{2}^{\mathrm{new}}(\Gamma_0(40))^{\oplus 6} โŠ•\oplus S2new(ฮ“0(48))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(48))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(80))โŠ•4S_{2}^{\mathrm{new}}(\Gamma_0(80))^{\oplus 4} โŠ•\oplus S2new(ฮ“0(96))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(96))^{\oplus 2} โŠ•\oplus S2new(ฮ“0(120))โŠ•3S_{2}^{\mathrm{new}}(\Gamma_0(120))^{\oplus 3} โŠ•\oplus S2new(ฮ“0(160))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(160))^{\oplus 2} โŠ•\oplus S2new(ฮ“0(240))โŠ•2S_{2}^{\mathrm{new}}(\Gamma_0(240))^{\oplus 2}

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.481 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, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 80, 96, 120, 160, 240, 480]: [0, 1, 1, 2, 1, 3, 3, 3, 5, 4, 4, 6, 8, 10, 6, 10, 12, 18, 16, 20, 32, 28, 56, 104] (certified at ell = 2147483497)
  • T = [(7, 1), (11, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12), (53, 13), (59, 14), (61, 15), (67, 16)]; 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(480, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 480:2 1000

โ† 479.2.a ยท 481.2.a โ†’