Atlas › Modular forms › Weight 2 › Level 501 › 501.2.a

Newspace 501.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(501))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
501.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - 4q^{5} - q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$	501a1 rank 0
501.2.a.b	5	degree 5	$5q - 4q^{2} + 5q^{3} + 4q^{4} - 9q^{5} - 4q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
501.2.a.c	5	degree 5	$5q - 5q^{3} - q^{5} - 4q^{7} + O(q^{8})$	+ +	$-1$
501.2.a.d	8	degree 8	$8q - 3q^{2} - 8q^{3} + 13q^{4} + q^{5} + 3q^{6} + O(q^{8})$	+ −	$+1$
501.2.a.e	8	degree 8	$8q + 3q^{2} + 8q^{3} + 9q^{4} + 7q^{5} + 3q^{6} - 4q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
501.2.a.a	$x + 3$
501.2.a.b	$x^{5} + 13x^{4} + 57x^{3} + 92x^{2} + 25x - 25$
501.2.a.c	$x^{5} + x^{4} - 5x^{3} - 2x^{2} + 5x + 1$
501.2.a.d	$x^{8} + 2x^{7} - 41x^{6} - 31x^{5} + 531x^{4} - 15x^{3} - 2310x^{2} + 1210x + 1219$
501.2.a.e	$x^{8} - 10x^{7} + 21x^{6} + 69x^{5} - 305x^{4} + 303x^{3} + 122x^{2} - 336x + 133$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(501))$	27
Old forms	28
Cusp forms $S_{2}(\Gamma_0(501))$	55
Eisenstein series $E_{2}(\Gamma_0(501))$	3
Modular forms $M_{2}(\Gamma_0(501))$	58

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(501)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(167))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.293 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, 3, 167, 501]: [0, 1, 15, 58] (certified at ell = 2147483237)
T = [(2, 1), (5, 1)]; h of degree 27 = 1 x 27 (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(501, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 501:2 1000

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