Atlas › Modular forms › Weight 2 › Level 976 › 976.2.a

Newspace 976.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(976))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
976.2.a.a	1	$\mathbb{Q}$	$q - 3q^{5} + 3q^{7} + O(q^{8})$	−	976c1 rank 1
976.2.a.b	1	$\mathbb{Q}$	$q + 2q^{3} - 3q^{5} - q^{7} + O(q^{8})$	+	976b1 rank 0
976.2.a.c	1	$\mathbb{Q}$	$q + 2q^{3} + q^{5} + 5q^{7} + O(q^{8})$	+	976a1 rank 0
976.2.a.d	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{3} - 5q^{7} + O(q^{8})$	−
976.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{5} + 2q^{7} + O(q^{8})$	+
976.2.a.f	3	degree 3	$3q - 2q^{3} - q^{5} + 3q^{7} + O(q^{8})$	−
976.2.a.g	3	degree 3	$3q + q^{3} + q^{5} - 4q^{7} + O(q^{8})$	+
976.2.a.h	3	degree 3	$3q + 2q^{3} - q^{5} + 7q^{7} + O(q^{8})$	−
976.2.a.i	4	degree 4	$4q - q^{3} - 2q^{5} - 7q^{7} + O(q^{8})$	+
976.2.a.j	4	degree 4	$4q + 5q^{5} + q^{7} + O(q^{8})$	+
976.2.a.k	6	degree 6	$6q - 3q^{3} + 5q^{5} - 6q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{17} + 7T_{19} + 8T_{23}$ 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 $T$
976.2.a.a	$x + 38$
976.2.a.b	$x - 142$
976.2.a.c	$x + 24$
976.2.a.d	$x^{2} - 11x - 701$
976.2.a.e	$x^{2} + 64x + 574$
976.2.a.f	$x^{3} + 144x^{2} + 1844x - 271442$
976.2.a.g	$x^{3} + 17x^{2} - 9609x - 326042$
976.2.a.h	$x^{3} - 116x^{2} + 1958x + 9802$
976.2.a.i	$x^{4} + 209x^{3} + 12365x^{2} + 130096x - 2173582$
976.2.a.j	$x^{4} - 64x^{3} - 378x^{2} + 77002x - 1023168$
976.2.a.k	$x^{6} - 61x^{5} - 12109x^{4} + 945526x^{3} + 12221800x^{2} - 2023950030x + 30780766924$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(976))$	30
Old forms	89
Cusp forms $S_{2}(\Gamma_0(976))$	119
Eisenstein series $E_{2}(\Gamma_0(976))$	11
Modular forms $M_{2}(\Gamma_0(976))$	130

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(976)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(61))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(122))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(244))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(488))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.682 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, 4, 8, 16, 61, 122, 244, 488, 976]: [0, 1, 2, 3, 4, 5, 17, 34, 66, 128] (certified at ell = 2147482921)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (13, 5), (17, 6), (19, 7), (23, 8)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 9 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(976, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 976:2 1000

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