Atlas › Modular forms › Weight 2 › Level 963 › 963.2.a

Newspace 963.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(963))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
963.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - q^{4} - 2q^{5} - 4q^{7} + O(q^{8})$	−
963.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - q^{4} + 3q^{5} - 4q^{7} + O(q^{8})$	−
963.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - q^{4} + 6q^{5} - 2q^{7} + O(q^{8})$	+
963.2.a.d	6	degree 6	$6q - 3q^{2} + 7q^{4} - 6q^{5} + O(q^{8})$	−
963.2.a.e	7	degree 7	$7q + 14q^{4} + 8q^{5} + 6q^{7} + O(q^{8})$	+
963.2.a.f	7	degree 7	$7q + q^{2} + 7q^{4} - 5q^{5} + 4q^{7} + O(q^{8})$	+
963.2.a.g	9	degree 9	$9q - 3q^{2} + 9q^{4} - 14q^{5} + O(q^{8})$	+
963.2.a.h	9	degree 9	$9q + 3q^{2} + 9q^{4} + 14q^{5} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
963.2.a.a	$x^{2} + 5x + 5$
963.2.a.b	$x^{2} - 10x + 20$
963.2.a.c	$x^{2} - 19x + 89$
963.2.a.d	$x^{6} + 21x^{5} + 88x^{4} - 339x^{3} - 1394x^{2} + 1303x + 3117$
963.2.a.e	$x^{7} - 24x^{6} + 19x^{5} + 2605x^{4} - 9563x^{3} - 62819x^{2} + 168818x + 613639$
963.2.a.f	$x^{7} + 14x^{6} - 103x^{5} - 1730x^{4} + 1328x^{3} + 52848x^{2} + 80992x - 27616$
963.2.a.g	$x^{9} + 45x^{8} + 726x^{7} + 4144x^{6} - 13601x^{5} - 279053x^{4} - 1209766x^{3} - 1471266x^{2} + 2060857x + 4162663$
963.2.a.h	$x^{9} - 45x^{8} + 726x^{7} - 4144x^{6} - 13601x^{5} + 279053x^{4} - 1209766x^{3} + 1471266x^{2} + 2060857x - 4162663$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(963))$	44
Old forms	61
Cusp forms $S_{2}(\Gamma_0(963))$	105
Eisenstein series $E_{2}(\Gamma_0(963))$	7
Modular forms $M_{2}(\Gamma_0(963))$	112

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(963)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(107))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(321))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.393 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, 9, 107, 321, 963]: [0, 1, 2, 10, 38, 110] (certified at ell = 2147482621)
T = [(2, 1), (5, 3)]; h of degree 44 = 1 x 44 (phi(ord eps) x dim S_k^new); 7 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(963, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 963:2 1000

← 962.2.a · 964.2.a →