Atlas › Modular forms › Weight 2 › Level 724 › 724.2.a

Newspace 724.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(724))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
724.2.a.a	2	$\mathbb{Q}(\sqrt{13})$	$2q + 4q^{3} + 5q^{5} - 3q^{7} + O(q^{8})$	+
724.2.a.b	5	degree 5	$5q - q^{3} - 5q^{5} - 4q^{7} + O(q^{8})$	−
724.2.a.c	8	degree 8	$8q - 5q^{3} + 2q^{5} + 5q^{7} + O(q^{8})$	+

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 $T$
724.2.a.a	$x^{2} - 9x + 17$
724.2.a.b	$x^{5} + 6x^{4} + 5x^{3} - 9x^{2} - 5x + 3$
724.2.a.c	$x^{8} + 3x^{7} - 38x^{6} - 68x^{5} + 389x^{4} + 493x^{3} - 1138x^{2} - 1041x - 18$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(724))$	15
Old forms	74
Cusp forms $S_{2}(\Gamma_0(724))$	89
Eisenstein series $E_{2}(\Gamma_0(724))$	5
Modular forms $M_{2}(\Gamma_0(724))$	94

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(724)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(181))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(362))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.22 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, 181, 362, 724]: [0, 1, 2, 15, 47, 94] (certified at ell = 2147482801)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (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(724, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 724:2 1000

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