Atlas › Modular forms › Weight 2 › Level 764 › 764.2.a

Newspace 764.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(764))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
764.2.a.a	8	degree 8	$8q - 5q^{3} - 4q^{5} - 7q^{7} + O(q^{8})$	−
764.2.a.b	8	degree 8	$8q + 3q^{3} + 5q^{7} + O(q^{8})$	+

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 $T$
764.2.a.a	$x^{8} + 9x^{7} + 8x^{6} - 87x^{5} - 173x^{4} + 2x^{3} + 195x^{2} + 135x + 27$
764.2.a.b	$x^{8} - 3x^{7} - 44x^{6} + 113x^{5} + 523x^{4} - 1022x^{3} - 1753x^{2} + 1547x - 289$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(764))$	16
Old forms	78
Cusp forms $S_{2}(\Gamma_0(764))$	94
Eisenstein series $E_{2}(\Gamma_0(764))$	5
Modular forms $M_{2}(\Gamma_0(764))$	99

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(764)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(191))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(382))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.222 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, 191, 382, 764]: [0, 1, 2, 17, 50, 99] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 16 = 1 x 16 (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(764, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 764:2 1000

← 763.2.a · 765.2.a →