Atlas › Modular forms › Weight 2 › Level 776 › 776.2.a

Newspace 776.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(776))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
776.2.a.a	1	$\mathbb{Q}$	$q - 2q^{5} + 2q^{7} + O(q^{8})$	−	776a1 rank 1
776.2.a.b	3	degree 3	$3q - 4q^{3} + 3q^{5} - 5q^{7} + O(q^{8})$	−
776.2.a.c	3	degree 3	$3q - 3q^{5} - 3q^{7} + O(q^{8})$	+
776.2.a.d	8	degree 8	$8q + 6q^{3} - q^{5} + 3q^{7} + O(q^{8})$	+
776.2.a.e	9	degree 9	$9q - 2q^{3} + 5q^{5} + 3q^{7} + O(q^{8})$	−

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 $T$
776.2.a.a	$x + 2$
776.2.a.b	$x^{3} + x^{2} - 10x + 9$
776.2.a.c	$x^{3} + 3x^{2} - 1$
776.2.a.d	$x^{8} - 5x^{7} - 40x^{6} + 199x^{5} + 432x^{4} - 2128x^{3} - 1392x^{2} + 5740x + 2512$
776.2.a.e	$x^{9} - 3x^{8} - 60x^{7} + 179x^{6} + 760x^{5} - 2142x^{4} - 2712x^{3} + 6340x^{2} + 2304x + 144$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(776))$	24
Old forms	71
Cusp forms $S_{2}(\Gamma_0(776))$	95
Eisenstein series $E_{2}(\Gamma_0(776))$	7
Modular forms $M_{2}(\Gamma_0(776))$	102

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(776)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(97))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(194))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(388))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.245 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, 8, 97, 194, 388, 776]: [0, 1, 2, 3, 8, 26, 52, 102] (certified at ell = 2147483137)
T = [(3, 1), (5, 1)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 3 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(776, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 776:2 1000

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