Atlas › Modular forms › Weight 2 › Level 778 › 778.2.a

Newspace 778.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(778))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
778.2.a.a	4	degree 4	$4q + 4q^{2} - 3q^{3} + 4q^{4} - 7q^{5} - 3q^{6} - 5q^{7} + O(q^{8})$	− −	$-1$
778.2.a.b	7	degree 7	$7q - 7q^{2} + 5q^{3} + 7q^{4} + 9q^{5} - 5q^{6} + 3q^{7} + O(q^{8})$	+ −	$+1$
778.2.a.c	9	degree 9	$9q - 9q^{2} - 6q^{3} + 9q^{4} - 8q^{5} + 6q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$
778.2.a.d	12	degree 12	$12q + 12q^{2} + 4q^{3} + 12q^{4} + 12q^{5} + 4q^{6} + 5q^{7} + O(q^{8})$	− +	$+1$

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 $T$
778.2.a.a	$x^{4} + 24x^{3} + 191x^{2} + 594x + 571$
778.2.a.b	$x^{7} - 32x^{6} + 279x^{5} + 330x^{4} - 15433x^{3} + 74252x^{2} - 129452x + 61464$
778.2.a.c	$x^{9} + 30x^{8} + 186x^{7} - 1868x^{6} - 22216x^{5} - 14969x^{4} + 450929x^{3} + 1069793x^{2} - 1015510x - 1662199$
778.2.a.d	$x^{12} - 40x^{11} + 412x^{10} + 3380x^{9} - 88960x^{8} + 338007x^{7} + 3537033x^{6} - 30529913x^{5} + 22780072x^{4} + 424531629x^{3} - 1341679812x^{2} + 1422796884x - 500793320$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(778))$	32
Old forms	64
Cusp forms $S_{2}(\Gamma_0(778))$	96
Eisenstein series $E_{2}(\Gamma_0(778))$	3
Modular forms $M_{2}(\Gamma_0(778))$	99

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(778)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(389))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.248 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, 389, 778]: [0, 1, 33, 99] (certified at ell = 2147479897)
T = [(3, 1), (5, 3)]; h of degree 32 = 1 x 32 (phi(ord eps) x dim S_k^new); 5 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(778, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 778:2 1000

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