Atlas › Modular forms › Weight 2 › Level 772 › 772.2.a

Newspace 772.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(772))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
772.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 4q^{5} + q^{7} + O(q^{8})$	−
772.2.a.b	5	degree 5	$5q - 3q^{3} - 6q^{5} + O(q^{8})$	−
772.2.a.c	9	degree 9	$9q + 5q^{3} + 12q^{5} + q^{7} + O(q^{8})$	+

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 $T$
772.2.a.a	$x^{2} + 12x + 16$
772.2.a.b	$x^{5} + 21x^{4} + 57x^{3} - 554x^{2} - 2413x - 1091$
772.2.a.c	$x^{9} - 41x^{8} + 505x^{7} + 380x^{6} - 52533x^{5} + 334213x^{4} + 15516x^{3} - 5070568x^{2} + 5005696x + 23946128$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(772))$	16
Old forms	79
Cusp forms $S_{2}(\Gamma_0(772))$	95
Eisenstein series $E_{2}(\Gamma_0(772))$	5
Modular forms $M_{2}(\Gamma_0(772))$	100

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(772)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(193))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(386))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.238 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, 193, 386, 772]: [0, 1, 2, 16, 50, 100] (certified at ell = 2147483137)
T = [(3, 1), (5, 3)]; h of degree 16 = 1 x 16 (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(772, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 772:2 1000

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