Atlas › Modular forms › Weight 2 › Level 508 › 508.2.a

Newspace 508.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(508))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
508.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} - 4q^{5} + O(q^{8})$	−
508.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q + 4q^{3} + 2q^{5} - 2q^{7} + O(q^{8})$	+
508.2.a.c	3	degree 3	$3q - 3q^{3} - 3q^{7} + O(q^{8})$	−
508.2.a.d	3	degree 3	$3q + q^{3} + 4q^{5} + 7q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
508.2.a.a	$x^{2} + 6x + 4$
508.2.a.b	$x^{2} - 6x + 4$
508.2.a.c	$x^{3} + 3x^{2} - 1$
508.2.a.d	$x^{3} - 5x^{2} - 4x + 29$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(508))$	10
Old forms	52
Cusp forms $S_{2}(\Gamma_0(508))$	62
Eisenstein series $E_{2}(\Gamma_0(508))$	5
Modular forms $M_{2}(\Gamma_0(508))$	67

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(508)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(127))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(254))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.201 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, 127, 254, 508]: [0, 1, 2, 11, 34, 67] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 10 = 1 x 10 (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(508, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 508:2 1000

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