Atlas › Modular forms › Weight 2 › Level 596 › 596.2.a

Newspace 596.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(596))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
596.2.a.a	3	degree 3	$3q - 3q^{5} - 3q^{7} + O(q^{8})$	−
596.2.a.b	10	degree 10	$10q + 5q^{5} + 5q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
596.2.a.a	$x^{3} + 9x^{2} + 15x - 1$
596.2.a.b	$x^{10} - 15x^{9} - 285x^{8} + 5297x^{7} + 13175x^{6} - 557877x^{5} + 1668398x^{4} + 14337852x^{3} - 95211225x^{2} + 169851491x - 49292028$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(596))$	13
Old forms	60
Cusp forms $S_{2}(\Gamma_0(596))$	73
Eisenstein series $E_{2}(\Gamma_0(596))$	5
Modular forms $M_{2}(\Gamma_0(596))$	78

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(596)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(149))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(298))^{\oplus 2}$

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, 149, 298, 596]: [0, 1, 2, 13, 39, 78] (certified at ell = 2147481629)
T = [(3, 1), (5, 3)]; 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(596, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 596:2 1000

← 595.2.a · 597.2.a →