Atlas › Modular forms › Weight 10 › Level 37 › 37.10.a

Newspace 37.10.a: $S_{10}^{\mathrm{new}}(\Gamma_0(37))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
37.10.a.a	13	degree 13	$13q - 32q^{2} - 251q^{3} + 2730q^{4} - 2159q^{5} - 4401q^{6} - 12576q^{7} + O(q^{8})$	+	$-1$
37.10.a.b	14	degree 14	$14q + 48q^{2} + 397q^{3} + 3498q^{4} + 2841q^{5} + 3375q^{6} + 6632q^{7} + O(q^{8})$	−	$+1$

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 $T$
37.10.a.a	$x^{13} + 283x^{12} - 122743x^{11} - 31998132x^{10} + 6162080219x^{9} + 1305251766605x^{8} - 156366686581706x^{7} - 22991892854515203x^{6} + 1893737441952789659x^{5} + 173141238255369566084x^{4} - 9110126287645159038079x^{3} - 431085697660864288503321x^{2} + 10455196139397586329153065x - 54227391723920659885835900$
37.10.a.b	$x^{14} - 445x^{13} - 98435x^{12} + 60711620x^{11} + 2095424449x^{10} - 3061811325289x^{9} + 58838973622328x^{8} + 69130440034581907x^{7} - 2361980083380473937x^{6} - 676670439183284956320x^{5} + 15748170964979492905215x^{4} + 2574707559298497716523141x^{3} + 2046694429691111350155051x^{2} - 603091703766392998410384054x - 531641677049297363230589232$

Dimensions

Newforms $S_{10}^{\mathrm{new}}(\Gamma_0(37))$	27
Old forms	0
Cusp forms $S_{10}(\Gamma_0(37))$	27
Eisenstein series $E_{10}(\Gamma_0(37))$	2
Modular forms $M_{10}(\Gamma_0(37))$	29

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.315 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, 37]: [1, 29] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 8 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(37, 10)

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

cargo run --release -p sagebrush-web --example atlas -- 37:10 1000

← 36.10.a · 38.10.a →