Atlas › Modular forms › Weight 2 › Level 958 › 958.2.a

Newspace 958.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(958))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
958.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} - q^{3} + 2q^{4} + 5q^{5} + q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$
958.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 3q^{3} + 2q^{4} - q^{5} - 3q^{6} + O(q^{8})$	− −	$-1$
958.2.a.c	6	degree 6	$6q + 6q^{2} - 5q^{3} + 6q^{4} - 6q^{5} - 5q^{6} - 10q^{7} + O(q^{8})$	− −	$-1$
958.2.a.d	8	degree 8	$8q - 8q^{2} - 2q^{3} + 8q^{4} - 3q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
958.2.a.e	10	degree 10	$10q - 10q^{2} + 3q^{3} + 10q^{4} - 3q^{6} - 2q^{7} + O(q^{8})$	+ −	$+1$
958.2.a.f	11	degree 11	$11q + 11q^{2} + 4q^{3} + 11q^{4} + 3q^{5} + 4q^{6} + 6q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
958.2.a.a	$x^{2} - 4x - 1$
958.2.a.b	$x^{2} + 4x - 1$
958.2.a.c	$x^{6} + 11x^{5} + 40x^{4} + 40x^{3} - 68x^{2} - 156x - 73$
958.2.a.d	$x^{8} + 5x^{7} - 17x^{6} - 61x^{5} + 76x^{4} + 158x^{3} - 71x^{2} - 110x - 25$
958.2.a.e	$x^{10} - 3x^{9} - 52x^{8} + 140x^{7} + 968x^{6} - 2290x^{5} - 7731x^{4} + 15560x^{3} + 23888x^{2} - 37056x - 11952$
958.2.a.f	$x^{11} - 7x^{10} - 31x^{9} + 311x^{8} - 122x^{7} - 3348x^{6} + 5879x^{5} + 7054x^{4} - 19863x^{3} + 2694x^{2} + 8524x + 1000$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(958))$	39
Old forms	80
Cusp forms $S_{2}(\Gamma_0(958))$	119
Eisenstein series $E_{2}(\Gamma_0(958))$	3
Modular forms $M_{2}(\Gamma_0(958))$	122

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(958)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(479))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.379 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, 2, 479, 958]: [0, 1, 41, 122] (certified at ell = 2147482877)
T = [(3, 1), (5, 1)]; h of degree 39 = 1 x 39 (phi(ord eps) x dim S_k^new); 5 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(958, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 958:2 1000

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