Atlas › Modular forms › Weight 2 › Level 580 › 580.2.a

Newspace 580.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(580))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
580.2.a.a	1	$\mathbb{Q}$	$q - q^{5} + O(q^{8})$	+ −	580a1 rank 1
580.2.a.b	1	$\mathbb{Q}$	$q + q^{5} - 2q^{7} + O(q^{8})$	− +	580b1 rank 1
580.2.a.c	3	degree 3	$3q + 2q^{3} - 3q^{5} - 4q^{7} + O(q^{8})$	+ +
580.2.a.d	3	degree 3	$3q + 2q^{3} + 3q^{5} + 2q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
580.2.a.a	$x + 24$
580.2.a.b	$x + 104$
580.2.a.c	$x^{3} - 34x^{2} - 5076x + 19176$
580.2.a.d	$x^{3} - 122x^{2} + 4236x - 34840$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(580))$	8
Old forms	77
Cusp forms $S_{2}(\Gamma_0(580))$	85
Eisenstein series $E_{2}(\Gamma_0(580))$	11
Modular forms $M_{2}(\Gamma_0(580))$	96

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(580)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(58))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(116))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(145))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(290))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.4 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, 5, 10, 20, 29, 58, 116, 145, 290, 580]: [0, 1, 2, 1, 3, 6, 3, 9, 18, 16, 48, 96] (certified at ell = 2147483549)
T = [(3, 1), (7, 3), (11, 5), (13, 7), (17, 9)]; h of degree 8 = 1 x 8 (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(580, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 580:2 1000

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