Atlas › Modular forms › Weight 2 › Level 668 › 668.2.a

Newspace 668.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(668))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
668.2.a.a	2	$\mathbb{Q}(\sqrt{13})$	$2q + q^{3} - 6q^{5} + 3q^{7} + O(q^{8})$	+
668.2.a.b	5	degree 5	$5q + 3q^{3} + 10q^{5} + 9q^{7} + O(q^{8})$	+
668.2.a.c	7	degree 7	$7q - 4q^{3} - 2q^{5} - 12q^{7} + O(q^{8})$	−

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 $T$
668.2.a.a	$x^{2} + 5x + 3$
668.2.a.b	$x^{5} - 13x^{4} + 58x^{3} - 99x^{2} + 45x - 5$
668.2.a.c	$x^{7} + 6x^{6} + 2x^{5} - 44x^{4} - 94x^{3} - 69x^{2} - 18x - 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(668))$	14
Old forms	68
Cusp forms $S_{2}(\Gamma_0(668))$	82
Eisenstein series $E_{2}(\Gamma_0(668))$	5
Modular forms $M_{2}(\Gamma_0(668))$	87

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(668)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(167))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(334))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.214 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, 167, 334, 668]: [0, 1, 2, 15, 44, 87] (certified at ell = 2147483237)
T = [(3, 1), (5, 1)]; h of degree 14 = 1 x 14 (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(668, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 668:2 1000

← 667.2.a · 669.2.a →