๐ŸŒฟ Sagebrush Atlas
Atlas โ€บ Modular forms โ€บ Weight 4 โ€บ Level 216 โ€บ 216.4.a

Newspace 216.4.a: S4new(ฮ“0(216))S_{4}^{\mathrm{new}}(\Gamma_0(216))

Newforms

LabelDimCoefficient fieldTrace formAL signsSignCMElliptic curve
216.4.a.a1Q\mathbb{Q} qโˆ’4q5+3q7+O(q8)q - 4q^{5} + 3q^{7} + O(q^{8})
216.4.a.b1Q\mathbb{Q} qโˆ’q5โˆ’9q7+O(q8)q - q^{5} - 9q^{7} + O(q^{8})
216.4.a.c1Q\mathbb{Q} q+q5โˆ’9q7+O(q8)q + q^{5} - 9q^{7} + O(q^{8})
216.4.a.d1Q\mathbb{Q} q+4q5+3q7+O(q8)q + 4q^{5} + 3q^{7} + O(q^{8})
216.4.a.e2Q(33)\mathbb{Q}(\sqrt{33}) 2qโˆ’8q5+24q7+O(q8)2q - 8q^{5} + 24q^{7} + O(q^{8})
216.4.a.f2Q(5)\mathbb{Q}(\sqrt{5}) 2qโˆ’4q5โˆ’6q7+O(q8)2q - 4q^{5} - 6q^{7} + O(q^{8})
216.4.a.g2Q(5)\mathbb{Q}(\sqrt{5}) 2q+4q5โˆ’6q7+O(q8)2q + 4q^{5} - 6q^{7} + O(q^{8})
216.4.a.h2Q(33)\mathbb{Q}(\sqrt{33}) 2q+8q5+24q7+O(q8)2q + 8q^{5} + 24q^{7} + O(q^{8})

Hecke characteristic polynomials

The eigenvalue of T=T5+3T7T = T_{5} + 3T_{7} 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
216.4.a.axโˆ’5x - 5
216.4.a.bx+28x + 28
216.4.a.cx+26x + 26
216.4.a.dxโˆ’13x - 13
216.4.a.ex2โˆ’64xโˆ’164x^{2} - 64x - 164
216.4.a.fx2+22xโˆ’4379x^{2} + 22x - 4379
216.4.a.gx2+14xโˆ’8771x^{2} + 14x - 8771
216.4.a.hx2โˆ’80xโˆ’3152x^{2} - 80x - 3152

Dimensions

Newforms S4new(ฮ“0(216))S_{4}^{\mathrm{new}}(\Gamma_0(216))12
Old forms84
Cusp forms S4(ฮ“0(216))S_{4}(\Gamma_0(216))96
Eisenstein series E4(ฮ“0(216))E_{4}(\Gamma_0(216))24
Modular forms M4(ฮ“0(216))M_{4}(\Gamma_0(216))120

Decomposition of the old subspace

S4old(ฮ“0(216))=S_{4}^{\mathrm{old}}(\Gamma_0(216)) = S4new(ฮ“0(6))โŠ•9S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 9} โŠ•\oplus S4new(ฮ“0(8))โŠ•4S_{4}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 4} โŠ•\oplus S4new(ฮ“0(9))โŠ•8S_{4}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 8} โŠ•\oplus S4new(ฮ“0(12))โŠ•6S_{4}^{\mathrm{new}}(\Gamma_0(12))^{\oplus 6} โŠ•\oplus S4new(ฮ“0(18))โŠ•6S_{4}^{\mathrm{new}}(\Gamma_0(18))^{\oplus 6} โŠ•\oplus S4new(ฮ“0(24))โŠ•3S_{4}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 3} โŠ•\oplus S4new(ฮ“0(27))โŠ•4S_{4}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 4} โŠ•\oplus S4new(ฮ“0(36))โŠ•4S_{4}^{\mathrm{new}}(\Gamma_0(36))^{\oplus 4} โŠ•\oplus S4new(ฮ“0(54))โŠ•3S_{4}^{\mathrm{new}}(\Gamma_0(54))^{\oplus 3} โŠ•\oplus S4new(ฮ“0(72))โŠ•2S_{4}^{\mathrm{new}}(\Gamma_0(72))^{\oplus 2} โŠ•\oplus S4new(ฮ“0(108))โŠ•2S_{4}^{\mathrm{new}}(\Gamma_0(108))^{\oplus 2}

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.202 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, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216]: [1, 2, 2, 3, 5, 5, 4, 9, 11, 16, 10, 21, 29, 40, 57, 112] (certified at ell = 2147483647)
  • T = [(5, 1), (7, 3)]; h of degree 12 = 1 x 12 (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(216, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 216:4 1000

โ† 215.4.a ยท 217.4.a โ†’