Atlas › Modular forms › Weight 2 › Level 766 › 766.2.a

Newspace 766.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(766))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
766.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} + 2q^{5} - 2q^{6} + 3q^{7} + O(q^{8})$	+ −	$+1$	766a1 rank 0
766.2.a.b	4	degree 4	$4q + 4q^{2} + 8q^{3} + 4q^{4} + 2q^{5} + 8q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$
766.2.a.c	5	degree 5	$5q - 5q^{2} - 2q^{3} + 5q^{4} - 3q^{5} + 2q^{6} + 3q^{7} + O(q^{8})$	+ +	$-1$
766.2.a.d	5	degree 5	$5q + 5q^{2} - 4q^{3} + 5q^{4} - 5q^{5} - 4q^{6} - 9q^{7} + O(q^{8})$	− −	$-1$
766.2.a.e	6	degree 6	$6q + 6q^{2} - 4q^{3} + 6q^{4} + q^{5} - 4q^{6} + 3q^{7} + O(q^{8})$	− +	$+1$
766.2.a.f	10	degree 10	$10q - 10q^{2} + 10q^{4} + 3q^{5} - 12q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
766.2.a.a	$x - 4$
766.2.a.b	$x^{4} - 10x^{3} + 28x^{2} - 16x - 4$
766.2.a.c	$x^{5} + 5x^{4} - 29x^{2} - 29x + 11$
766.2.a.d	$x^{5} + 9x^{4} + 28x^{3} + 35x^{2} + 15x + 1$
766.2.a.e	$x^{6} + 3x^{5} - 28x^{4} - 35x^{3} + 225x^{2} - 125x + 6$
766.2.a.f	$x^{10} - 3x^{9} - 64x^{8} + 111x^{7} + 1223x^{6} - 943x^{5} - 7486x^{4} - 1282x^{3} + 7328x^{2} + 4872x + 872$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(766))$	31
Old forms	64
Cusp forms $S_{2}(\Gamma_0(766))$	95
Eisenstein series $E_{2}(\Gamma_0(766))$	3
Modular forms $M_{2}(\Gamma_0(766))$	98

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(766)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(383))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.243 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, 383, 766]: [0, 1, 33, 98] (certified at ell = 2147476031)
T = [(3, 1), (5, 1)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 4 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(766, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 766:2 1000

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