Atlas › Modular forms › Weight 2 › Level 581 › 581.2.a

Newspace 581.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(581))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
581.2.a.a	7	degree 7	$7q - 2q^{2} - 3q^{3} + 2q^{4} - q^{5} - 2q^{6} - 7q^{7} + O(q^{8})$	+ +	$-1$
581.2.a.b	7	degree 7	$7q - 7q^{3} + 2q^{4} - 5q^{5} - 4q^{6} + 7q^{7} + O(q^{8})$	− −	$-1$
581.2.a.c	13	degree 13	$13q + 7q^{3} + 16q^{4} + 5q^{5} + 13q^{7} + O(q^{8})$	− +	$+1$
581.2.a.d	14	degree 14	$14q + q^{2} + 3q^{3} + 23q^{4} - 5q^{5} + 6q^{6} - 14q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
581.2.a.a	$x^{7} + 5x^{6} - 5x^{5} - 38x^{4} - 6x^{3} + 17x^{2} + 8x + 1$
581.2.a.b	$x^{7} + 7x^{6} + 7x^{5} - 30x^{4} - 36x^{3} + 39x^{2} + 24x - 13$
581.2.a.c	$x^{13} - 7x^{12} - 23x^{11} + 256x^{10} - 207x^{9} - 2060x^{8} + 3752x^{7} + 4890x^{6} - 14045x^{5} - 33x^{4} + 14708x^{3} - 3586x^{2} - 3644x - 413$
581.2.a.d	$x^{14} - 4x^{13} - 56x^{12} + 205x^{11} + 1155x^{10} - 3469x^{9} - 12416x^{8} + 25206x^{7} + 74851x^{6} - 72970x^{5} - 230505x^{4} + 25616x^{3} + 257846x^{2} + 115377x + 12929$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(581))$	41
Old forms	14
Cusp forms $S_{2}(\Gamma_0(581))$	55
Eisenstein series $E_{2}(\Gamma_0(581))$	3
Modular forms $M_{2}(\Gamma_0(581))$	58

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(581)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(83))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.298 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, 7, 83, 581]: [0, 1, 8, 58] (certified at ell = 2147483323)
T = [(2, 1), (3, 1)]; h of degree 41 = 1 x 41 (phi(ord eps) x dim S_k^new); 4 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(581, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 581:2 1000

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