Atlas › Modular forms › Weight 2 › Level 308 › 308.2.a

Newspace 308.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(308))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
308.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - q^{5} - q^{7} + O(q^{8})$	+ −	308a1 rank 1
308.2.a.b	2	$\mathbb{Q}(\sqrt{6})$	$2q + 4q^{5} - 2q^{7} + O(q^{8})$	+ +
308.2.a.c	3	degree 3	$3q - q^{3} - q^{5} + 3q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31}$ 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 $T$
308.2.a.a	$x + 37$
308.2.a.b	$x^{2} - 120x + 3384$
308.2.a.c	$x^{3} + 41x^{2} - 2776x + 12336$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(308))$	6
Old forms	37
Cusp forms $S_{2}(\Gamma_0(308))$	43
Eisenstein series $E_{2}(\Gamma_0(308))$	11
Modular forms $M_{2}(\Gamma_0(308))$	54

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(308)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(44))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(77))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(154))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.269 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, 7, 11, 14, 22, 28, 44, 77, 154, 308]: [0, 1, 2, 1, 2, 4, 5, 7, 9, 10, 28, 54] (certified at ell = 2147482951)
T = [(3, 1), (5, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8)]; 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(308, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 308:2 1000

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