Atlas › Modular forms › Weight 2 › Level 628 › 628.2.a

Newspace 628.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(628))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
628.2.a.a	1	$\mathbb{Q}$	$q + 2q^{3} + 4q^{5} - q^{7} + O(q^{8})$	+	628a1 rank 0
628.2.a.b	5	degree 5	$5q - q^{3} - 5q^{5} - 5q^{7} + O(q^{8})$	−
628.2.a.c	7	degree 7	$7q - 3q^{3} - q^{5} + 2q^{7} + O(q^{8})$	+

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 $T$
628.2.a.a	$x - 6$
628.2.a.b	$x^{5} + 6x^{4} + 9x^{3} - 2x^{2} - 6x + 1$
628.2.a.c	$x^{7} + 4x^{6} - 25x^{5} - 34x^{4} + 248x^{3} - 345x^{2} + 188x - 36$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(628))$	13
Old forms	64
Cusp forms $S_{2}(\Gamma_0(628))$	77
Eisenstein series $E_{2}(\Gamma_0(628))$	5
Modular forms $M_{2}(\Gamma_0(628))$	82

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(628)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(157))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(314))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.211 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, 157, 314, 628]: [0, 1, 2, 13, 41, 82] (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(628, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 628:2 1000

← 627.2.a · 629.2.a →