Atlas › Modular forms › Weight 2 › Level 502 › 502.2.a

Newspace 502.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(502))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
502.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 3q^{5} - 2q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
502.2.a.b	2	$\mathbb{Q}(\sqrt{13})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 5q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	− +	$+1$
502.2.a.c	5	degree 5	$5q - 5q^{2} - q^{3} + 5q^{4} - 6q^{5} + q^{6} - q^{7} + O(q^{8})$	+ +	$-1$
502.2.a.d	5	degree 5	$5q - 5q^{2} + 2q^{3} + 5q^{4} + 7q^{5} - 2q^{6} + q^{7} + O(q^{8})$	+ −	$+1$
502.2.a.e	6	degree 6	$6q + 6q^{2} + q^{3} + 6q^{4} - q^{5} + q^{6} + 6q^{7} + O(q^{8})$	− +	$+1$

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 $T$
502.2.a.a	$x^{2} + 5x + 5$
502.2.a.b	$x^{2} - 7x + 9$
502.2.a.c	$x^{5} + 7x^{4} + 10x^{3} - 2x^{2} - 4x + 1$
502.2.a.d	$x^{5} - 9x^{4} + 11x^{3} + 78x^{2} - 208x + 136$
502.2.a.e	$x^{6} - 33x^{4} + 55x^{3} + 46x^{2} - 88x + 8$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(502))$	20
Old forms	42
Cusp forms $S_{2}(\Gamma_0(502))$	62
Eisenstein series $E_{2}(\Gamma_0(502))$	3
Modular forms $M_{2}(\Gamma_0(502))$	65

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(502)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(251))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.295 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, 251, 502]: [0, 1, 22, 65] (certified at ell = 2147482501)
T = [(3, 1), (5, 1)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 3 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(502, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 502:2 1000

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