Atlas › Modular forms › Weight 2 › Level 631 › 631.2.a

Newspace 631.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(631))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
631.2.a.a	20	degree 20	$20q - 8q^{2} - 4q^{3} + 14q^{4} - 17q^{5} - 11q^{6} - 6q^{7} + O(q^{8})$	+	$-1$
631.2.a.b	32	degree 32	$32q + 7q^{2} + 2q^{3} + 33q^{4} + 17q^{5} + 11q^{6} - 2q^{7} + O(q^{8})$	−	$+1$

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 $T$
631.2.a.a	$x^{20} + 12x^{19} + 26x^{18} - 203x^{17} - 896x^{16} + 793x^{15} + 8320x^{14} + 2525x^{13} - 36791x^{12} - 26373x^{11} + 87901x^{10} + 73605x^{9} - 113797x^{8} - 90414x^{7} + 70987x^{6} + 48258x^{5} - 13363x^{4} - 9607x^{3} - 1385x^{2} - 67x - 1$
631.2.a.b	$x^{32} - 9x^{31} - 93x^{30} + 1022x^{29} + 3122x^{28} - 48925x^{27} - 36651x^{26} + 1313568x^{25} - 400703x^{24} - 22224534x^{23} + 18740853x^{22} + 252376990x^{21} - 277215099x^{20} - 1998633274x^{19} + 2318645073x^{18} + 11289254743x^{17} - 12083838607x^{16} - 45830558540x^{15} + 39550647138x^{14} + 132248465893x^{13} - 76830700853x^{12} - 261633414220x^{11} + 72171389579x^{10} + 330817104732x^{9} + 2522785569x^{8} - 235526517316x^{7} - 59194270830x^{6} + 69303667601x^{5} + 33287114129x^{4} + 1747936018x^{3} - 939383427x^{2} - 128572656x - 3567193$

Orbit

Characteristic polynomial of

T

631.2.a.a

x^{20} + 12x^{19} + 26x^{18} - 203x^{17} - 896x^{16} + 793x^{15} + 8320x^{14} + 2525x^{13} - 36791x^{12} - 26373x^{11} + 87901x^{10} + 73605x^{9} - 113797x^{8} - 90414x^{7} + 70987x^{6} + 48258x^{5} - 13363x^{4} - 9607x^{3} - 1385x^{2} - 67x - 1

631.2.a.b

x^{32} - 9x^{31} - 93x^{30} + 1022x^{29} + 3122x^{28} - 48925x^{27} - 36651x^{26} + 1313568x^{25} - 400703x^{24} - 22224534x^{23} + 18740853x^{22} + 252376990x^{21} - 277215099x^{20} - 1998633274x^{19} + 2318645073x^{18} + 11289254743x^{17} - 12083838607x^{16} - 45830558540x^{15} + 39550647138x^{14} + 132248465893x^{13} - 76830700853x^{12} - 261633414220x^{11} + 72171389579x^{10} + 330817104732x^{9} + 2522785569x^{8} - 235526517316x^{7} - 59194270830x^{6} + 69303667601x^{5} + 33287114129x^{4} + 1747936018x^{3} - 939383427x^{2} - 128572656x - 3567193

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(631))$	52
Old forms	0
Cusp forms $S_{2}(\Gamma_0(631))$	52
Eisenstein series $E_{2}(\Gamma_0(631))$	1
Modular forms $M_{2}(\Gamma_0(631))$	53

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.217 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, 631]: [0, 53] (certified at ell = 2147478481)
T = [(2, 1), (3, 1)]; h of degree 52 = 1 x 52 (phi(ord eps) x dim S_k^new); 6 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(631, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 631:2 1000

← 630.2.a · 632.2.a →