Atlas › Modular forms › Weight 2 › Level 951 › 951.2.a

Newspace 951.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(951))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
951.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - 2q^{3} - q^{4} - q^{5} - q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
951.2.a.b	5	degree 5	$5q - 2q^{2} - 5q^{3} + 2q^{4} + 2q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
951.2.a.c	7	degree 7	$7q - 5q^{2} + 7q^{3} + 5q^{4} - 7q^{5} - 5q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
951.2.a.d	19	degree 19	$19q + 8q^{2} + 19q^{3} + 24q^{4} + 11q^{5} + 8q^{6} + O(q^{8})$	− +	$+1$
951.2.a.e	20	degree 20	$20q + q^{2} - 20q^{3} + 27q^{4} - q^{5} - q^{6} + 8q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
951.2.a.a	$x^{2} - 5$
951.2.a.b	$x^{5} + 2x^{4} - 7x^{3} - x^{2} + 5x + 1$
951.2.a.c	$x^{7} + 12x^{6} + 42x^{5} + 9x^{4} - 170x^{3} - 184x^{2} + 117x + 169$
951.2.a.d	$x^{19} - 19x^{18} + 84x^{17} + 549x^{16} - 5695x^{15} + 6781x^{14} + 83510x^{13} - 282353x^{12} - 298117x^{11} + 2582625x^{10} - 1711469x^{9} - 8955346x^{8} + 13653737x^{7} + 8797900x^{6} - 26112850x^{5} + 5745370x^{4} + 11680980x^{3} - 4559872x^{2} - 1034059x + 427715$
951.2.a.e	$x^{20} - 121x^{18} - 13x^{17} + 5946x^{16} + 1358x^{15} - 155119x^{14} - 55341x^{13} + 2354402x^{12} + 1159498x^{11} - 21429036x^{10} - 13742981x^{9} + 115839357x^{8} + 94108831x^{7} - 350622252x^{6} - 358095742x^{5} + 499041030x^{4} + 668961780x^{3} - 141701047x^{2} - 426985218x - 137624741$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(951))$	53
Old forms	52
Cusp forms $S_{2}(\Gamma_0(951))$	105
Eisenstein series $E_{2}(\Gamma_0(951))$	3
Modular forms $M_{2}(\Gamma_0(951))$	108

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(951)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(317))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.277 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, 3, 317, 951]: [0, 1, 27, 108] (certified at ell = 2147479753)
T = [(2, 1), (5, 1)]; h of degree 53 = 1 x 53 (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(951, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 951:2 1000

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