Atlas › Modular forms › Weight 2 › Level 372 › 372.2.a

Newspace 372.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(372))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
372.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - q^{5} - q^{7} + O(q^{8})$	+ −	372a1 rank 1
372.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 3q^{5} - 5q^{7} + O(q^{8})$	− +	372d1 rank 1
372.2.a.c	1	$\mathbb{Q}$	$q + q^{3} - 2q^{5} + 4q^{7} + O(q^{8})$	− −	372b1 rank 0
372.2.a.d	1	$\mathbb{Q}$	$q + q^{3} + 3q^{5} - q^{7} + O(q^{8})$	− −	372c1 rank 0
372.2.a.e	2	$\mathbb{Q}(\sqrt{17})$	$2q - 2q^{3} + 3q^{5} - q^{7} + O(q^{8})$	+ +

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
372.2.a.a	$x + 118$
372.2.a.b	$x + 72$
372.2.a.c	$x - 24$
372.2.a.d	$x - 14$
372.2.a.e	$x^{2} - 124x + 2144$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(372))$	6
Old forms	53
Cusp forms $S_{2}(\Gamma_0(372))$	59
Eisenstein series $E_{2}(\Gamma_0(372))$	11
Modular forms $M_{2}(\Gamma_0(372))$	70

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(372)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(62))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(93))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(124))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(186))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.268 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, 3, 4, 6, 12, 31, 62, 93, 124, 186, 372]: [0, 1, 1, 2, 3, 5, 3, 10, 12, 19, 36, 70] (certified at ell = 2147482951)
T = [(5, 1), (7, 3), (11, 5), (13, 7), (17, 9)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 2 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(372, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 372:2 1000

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