Atlas › Modular forms › Weight 10 › Level 31 › 31.10.a

Newspace 31.10.a: $S_{10}^{\mathrm{new}}(\Gamma_0(31))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
31.10.a.a	10	degree 10	$10q - 33q^{2} - 8q^{3} + 2389q^{4} - 3297q^{5} - 7680q^{6} - 495q^{7} + O(q^{8})$	+	$-1$
31.10.a.b	13	degree 13	$13q + 31q^{2} + 154q^{3} + 3925q^{4} + 1703q^{5} + 96q^{6} + 9109q^{7} + O(q^{8})$	−	$+1$

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 $T$
31.10.a.a	$x^{10} + 41x^{9} - 113648x^{8} - 2005658x^{7} + 4382772552x^{6} - 18513936584x^{5} - 63709733855552x^{4} + 1422127897557504x^{3} + 253313193944964224x^{2} - 3379429521593077760x - 261104268679451346944$
31.10.a.b	$x^{13} - 185x^{12} - 175168x^{11} + 24768405x^{10} + 11936200685x^{9} - 1067764522666x^{8} - 385899998175146x^{7} + 13058720908609236x^{6} + 5519273579377736152x^{5} + 94272585720179250160x^{4} - 20770430323470869998432x^{3} - 814793548973530859035968x^{2} - 2322731702795533516667520x + 142557653977616216030697216$

Dimensions

Newforms $S_{10}^{\mathrm{new}}(\Gamma_0(31))$	23
Old forms	0
Cusp forms $S_{10}(\Gamma_0(31))$	23
Eisenstein series $E_{10}(\Gamma_0(31))$	2
Modular forms $M_{10}(\Gamma_0(31))$	25

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.31 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, 31]: [1, 25] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; h of degree 23 = 1 x 23 (phi(ord eps) x dim S_k^new); 7 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(31, 10)

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

cargo run --release -p sagebrush-web --example atlas -- 31:10 1000

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