Atlas › Modular forms › Weight 2 › Level 524 › 524.2.a

Newspace 524.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(524))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
524.2.a.a	1	$\mathbb{Q}$	$q + q^{3} - 2q^{5} - 3q^{7} + O(q^{8})$	−	524a1 rank 1
524.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{3} + q^{5} - q^{7} + O(q^{8})$	−
524.2.a.c	4	degree 4	$4q + q^{3} - 3q^{5} - q^{7} + O(q^{8})$	+
524.2.a.d	4	degree 4	$4q + 3q^{3} + 2q^{5} + 7q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{17} + 7T_{19} + 9T_{23}$ 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 $T$
524.2.a.a	$x + 49$
524.2.a.b	$x^{2} + 58x + 821$
524.2.a.c	$x^{4} - 222x^{3} + 17454x^{2} - 571058x + 6616377$
524.2.a.d	$x^{4} - 5x^{3} - 6227x^{2} - 120381x + 1754127$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(524))$	11
Old forms	53
Cusp forms $S_{2}(\Gamma_0(524))$	64
Eisenstein series $E_{2}(\Gamma_0(524))$	5
Modular forms $M_{2}(\Gamma_0(524))$	69

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(524)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(131))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(262))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.246 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, 131, 262, 524]: [0, 1, 2, 12, 35, 69] (certified at ell = 2147482481)
T = [(3, 1), (5, 3), (17, 5), (19, 7), (23, 9)]; h of degree 11 = 1 x 11 (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(524, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 524:2 1000

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