Atlas › Modular forms › Weight 2 › Level 584 › 584.2.a

Newspace 584.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(584))$

Newforms

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

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
584.2.a.a	$x^{2} + 8x + 11$
584.2.a.b	$x^{3} - 43x - 106$
584.2.a.c	$x^{6} - 2x^{5} - 193x^{4} + 628x^{3} + 8602x^{2} - 26278x - 70762$
584.2.a.d	$x^{7} - 12x^{6} - 217x^{5} + 3288x^{4} + 1714x^{3} - 136586x^{2} + 112352x + 1637128$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(584))$	18
Old forms	53
Cusp forms $S_{2}(\Gamma_0(584))$	71
Eisenstein series $E_{2}(\Gamma_0(584))$	7
Modular forms $M_{2}(\Gamma_0(584))$	78

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(584)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(73))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(146))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(292))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.318 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, 8, 73, 146, 292, 584]: [0, 1, 2, 3, 6, 20, 40, 78] (certified at ell = 2147482873)
T = [(3, 1), (5, 3)]; h of degree 18 = 1 x 18 (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(584, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 584:2 1000

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