Atlas › Modular forms › Weight 2 › Level 479 › 479.2.a

Newspace 479.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(479))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
479.2.a.a	8	degree 8	$8q - 2q^{2} - 3q^{3} - 4q^{5} - q^{6} - 4q^{7} + O(q^{8})$	+	$-1$
479.2.a.b	32	degree 32	$32q + 3q^{2} + 5q^{3} + 43q^{4} + 6q^{5} + 7q^{6} + 8q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
479.2.a.a	$x^{8} + 5x^{7} - 3x^{6} - 34x^{5} - 11x^{4} + 47x^{3} + 43x^{2} + 12x + 1$
479.2.a.b	$x^{32} - 8x^{31} - 104x^{30} + 957x^{29} + 4228x^{28} - 48577x^{27} - 81959x^{26} + 1392943x^{25} + 570303x^{24} - 25295422x^{23} + 7329678x^{22} + 308331082x^{21} - 214638469x^{20} - 2601650507x^{19} + 2406842138x^{18} + 15412369331x^{17} - 15806504776x^{16} - 64251441694x^{15} + 65109425841x^{14} + 187190452708x^{13} - 165826086624x^{12} - 375101029632x^{11} + 238890398341x^{10} + 499959702602x^{9} - 142211793688x^{8} - 403862086316x^{7} - 42021772396x^{6} + 142171527973x^{5} + 68617794259x^{4} + 9029625271x^{3} - 471701802x^{2} - 109995678x + 2550717$

Orbit

Characteristic polynomial of

T

479.2.a.a

x^{8} + 5x^{7} - 3x^{6} - 34x^{5} - 11x^{4} + 47x^{3} + 43x^{2} + 12x + 1

479.2.a.b

x^{32} - 8x^{31} - 104x^{30} + 957x^{29} + 4228x^{28} - 48577x^{27} - 81959x^{26} + 1392943x^{25} + 570303x^{24} - 25295422x^{23} + 7329678x^{22} + 308331082x^{21} - 214638469x^{20} - 2601650507x^{19} + 2406842138x^{18} + 15412369331x^{17} - 15806504776x^{16} - 64251441694x^{15} + 65109425841x^{14} + 187190452708x^{13} - 165826086624x^{12} - 375101029632x^{11} + 238890398341x^{10} + 499959702602x^{9} - 142211793688x^{8} - 403862086316x^{7} - 42021772396x^{6} + 142171527973x^{5} + 68617794259x^{4} + 9029625271x^{3} - 471701802x^{2} - 109995678x + 2550717

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(479))$	40
Old forms	0
Cusp forms $S_{2}(\Gamma_0(479))$	40
Eisenstein series $E_{2}(\Gamma_0(479))$	1
Modular forms $M_{2}(\Gamma_0(479))$	41

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.195 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 479]: [0, 41] (certified at ell = 2147482877)
T = [(2, 1), (3, 1)]; h of degree 40 = 1 x 40 (phi(ord eps) x dim S_k^new); 4 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(479, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 479:2 1000

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