Atlas › Modular forms › Weight 2 › Level 531 › 531.2.a

Newspace 531.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(531))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
531.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - q^{4} - 2q^{5} + q^{7} + O(q^{8})$	−
531.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - q^{4} - 7q^{7} + O(q^{8})$	−
531.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{2} + 3q^{4} + 6q^{5} - 7q^{7} + O(q^{8})$	+
531.2.a.d	3	degree 3	$3q + 2q^{4} + 2q^{5} + 9q^{7} + O(q^{8})$	+
531.2.a.e	5	degree 5	$5q - 3q^{2} + 5q^{4} - 8q^{5} + 2q^{7} + O(q^{8})$	+
531.2.a.f	5	degree 5	$5q + 8q^{4} - 2q^{5} + 2q^{7} + O(q^{8})$	+
531.2.a.g	5	degree 5	$5q + 3q^{2} + 5q^{4} + 8q^{5} + 2q^{7} + O(q^{8})$	−

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 $T$
531.2.a.a	$x^{2} + 3x + 1$
531.2.a.b	$x^{2} - x - 1$
531.2.a.c	$x^{2} - 9x + 19$
531.2.a.d	$x^{3} - 2x^{2} - 14x - 13$
531.2.a.e	$x^{5} + 11x^{4} + 39x^{3} + 39x^{2} - 33x - 53$
531.2.a.f	$x^{5} + 2x^{4} - 15x^{3} - 9x^{2} + 54x - 14$
531.2.a.g	$x^{5} - 11x^{4} + 39x^{3} - 39x^{2} - 33x + 53$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(531))$	24
Old forms	33
Cusp forms $S_{2}(\Gamma_0(531))$	57
Eisenstein series $E_{2}(\Gamma_0(531))$	7
Modular forms $M_{2}(\Gamma_0(531))$	64

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(531)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(59))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(177))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.207 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, 9, 59, 177, 531]: [0, 1, 2, 6, 22, 62] (certified at ell = 2147481901)
T = [(2, 1), (5, 1)]; h of degree 24 = 1 x 24 (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(531, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 531:2 1000

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