Atlas › Modular forms › Weight 2 › Level 676 › 676.2.a

Newspace 676.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(676))$

Newforms

Label	Dim	Coefficient field	Trace form	Elliptic curve
676.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} - 3q^{5} - 4q^{7} + O(q^{8})$	676c1 rank 0
676.2.a.b	1	$\mathbb{Q}$	$q - 2q^{3} + 3q^{5} + 4q^{7} + O(q^{8})$	676b1 rank 0
676.2.a.c	1	$\mathbb{Q}$	$q - 2q^{5} + 2q^{7} + O(q^{8})$	676a1 rank 0
676.2.a.d	1	$\mathbb{Q}$	$q + 3q^{3} - 2q^{5} - q^{7} + O(q^{8})$	676e1 rank 0
676.2.a.e	1	$\mathbb{Q}$	$q + 3q^{3} + 2q^{5} + q^{7} + O(q^{8})$	676d1 rank 0
676.2.a.f	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} + O(q^{8})$
676.2.a.g	3	degree 3	$3q - 8q^{5} + q^{7} + O(q^{8})$
676.2.a.h	3	degree 3	$3q + 8q^{5} - q^{7} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ 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 $T$
676.2.a.a	$x - 37$
676.2.a.b	$x - 49$
676.2.a.c	$x - 148$
676.2.a.d	$x - 34$
676.2.a.e	$x + 28$
676.2.a.f	$x^{2} + 218x + 11638$
676.2.a.g	$x^{3} + 98x^{2} - 959x - 30863$
676.2.a.h	$x^{3} - 24x^{2} - 25x + 41$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(676))$	13
Old forms	58
Cusp forms $S_{2}(\Gamma_0(676))$	71
Eisenstein series $E_{2}(\Gamma_0(676))$	41
Modular forms $M_{2}(\Gamma_0(676))$	112

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(676)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(52))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(169))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(338))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.374 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, 13, 26, 52, 169, 338, 676]: [0, 1, 2, 1, 5, 10, 15, 47, 94] (certified at ell = 2147483053)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 4 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(676, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 676:2 1000

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