Atlas › Modular forms › Weight 2 › Level 716 › 716.2.a

Newspace 716.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(716))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
716.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{3} - q^{5} + O(q^{8})$	−
716.2.a.b	3	degree 3	$3q - 6q^{7} + O(q^{8})$	−
716.2.a.c	10	degree 10	$10q + 3q^{3} - q^{5} + 10q^{7} + O(q^{8})$	+

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 $T$
716.2.a.a	$x^{2} + 4x - 1$
716.2.a.b	$x^{3} - 3x + 1$
716.2.a.c	$x^{10} - 2x^{9} - 62x^{8} + 163x^{7} + 1039x^{6} - 2846x^{5} - 6438x^{4} + 18346x^{3} + 9807x^{2} - 41229x + 19013$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(716))$	15
Old forms	73
Cusp forms $S_{2}(\Gamma_0(716))$	88
Eisenstein series $E_{2}(\Gamma_0(716))$	5
Modular forms $M_{2}(\Gamma_0(716))$	93

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(716)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(179))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(358))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.219 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, 179, 358, 716]: [0, 1, 2, 16, 47, 93] (certified at ell = 2147483137)
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(716, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 716:2 1000

← 715.2.a · 717.2.a →