Atlas › Modular forms › Weight 2 › Level 424 › 424.2.a

Newspace 424.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(424))$

Newforms

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

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
424.2.a.a	$x^{2} + 6x + 7$
424.2.a.b	$x^{3} + 3x^{2} - x - 5$
424.2.a.c	$x^{3} - 5x^{2} - 7x + 43$
424.2.a.d	$x^{5} - 6x^{4} - 22x^{3} + 134x^{2} - 149x + 46$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(424))$	13
Old forms	38
Cusp forms $S_{2}(\Gamma_0(424))$	51
Eisenstein series $E_{2}(\Gamma_0(424))$	7
Modular forms $M_{2}(\Gamma_0(424))$	58

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(424)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(53))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(106))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(212))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.199 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, 53, 106, 212, 424]: [0, 1, 2, 3, 5, 15, 30, 58] (certified at ell = 2147483053)
T = [(3, 1), (5, 1)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 2 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(424, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 424:2 1000

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