Atlas › Modular forms › Weight 2 › Level 647 › 647.2.a

Newspace 647.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(647))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
647.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{2} + 3q^{4} + q^{5} + 5q^{6} - 5q^{7} + O(q^{8})$	+	$-1$
647.2.a.b	14	degree 14	$14q - 2q^{2} - 4q^{3} + 4q^{4} - 5q^{5} - 9q^{6} - 9q^{7} + O(q^{8})$	+	$-1$
647.2.a.c	38	degree 38	$38q + 6q^{2} + 4q^{3} + 46q^{4} + 2q^{5} - 4q^{6} + 18q^{7} + O(q^{8})$	−	$+1$

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 $T$
647.2.a.a	$x^{2} + 3x - 9$
647.2.a.b	$x^{14} + 6x^{13} - 7x^{12} - 82x^{11} - 13x^{10} + 383x^{9} + 164x^{8} - 705x^{7} - 348x^{6} + 450x^{5} + 254x^{4} - 51x^{3} - 32x^{2} + 2x + 1$
647.2.a.c	$x^{38} - 10x^{37} - 91x^{36} + 1181x^{35} + 2787x^{34} - 60183x^{33} - 9653x^{32} + 1763836x^{31} - 1633165x^{30} - 33277067x^{29} + 52469613x^{28} + 427658135x^{27} - 860816885x^{26} - 3863183851x^{25} + 8970065066x^{24} + 25001501920x^{23} - 63383764479x^{22} - 117586315312x^{21} + 311317699661x^{20} + 407854106532x^{19} - 1070069477330x^{18} - 1060648301364x^{17} + 2561349545718x^{16} + 2092447503844x^{15} - 4199692805544x^{14} - 3109928506401x^{13} + 4568253385809x^{12} + 3357859294324x^{11} - 3110271085440x^{10} - 2468189785923x^{9} + 1176348942817x^{8} + 1125380062293x^{7} - 169598424907x^{6} - 275695129707x^{5} - 18069844445x^{4} + 26963727625x^{3} + 5623560745x^{2} + 1968270x - 34670547$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(647))$	54
Old forms	0
Cusp forms $S_{2}(\Gamma_0(647))$	54
Eisenstein series $E_{2}(\Gamma_0(647))$	1
Modular forms $M_{2}(\Gamma_0(647))$	55

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.225 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, 647]: [0, 55] (certified at ell = 2147482943)
T = [(2, 1), (3, 1)]; h of degree 54 = 1 x 54 (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(647, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 647:2 1000

← 646.2.a · 648.2.a →