Atlas › Modular forms › Weight 2 › Level 779 › 779.2.a

Newspace 779.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(779))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
779.2.a.a	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{5} - 4q^{6} + 2q^{7} + O(q^{8})$	− −	$-1$
779.2.a.b	8	degree 8	$8q - 2q^{2} - 3q^{3} + 6q^{4} - 6q^{5} + q^{6} - 7q^{7} + O(q^{8})$	− −	$-1$
779.2.a.c	11	degree 11	$11q - 3q^{3} + 4q^{4} - 5q^{5} - 7q^{6} - 8q^{7} + O(q^{8})$	+ +	$-1$
779.2.a.d	19	degree 19	$19q + 5q^{3} + 24q^{4} + 2q^{5} + 5q^{6} + q^{7} + O(q^{8})$	− +	$+1$
779.2.a.e	21	degree 21	$21q + q^{2} + q^{3} + 29q^{4} + 11q^{5} + 13q^{6} + 6q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
779.2.a.a	$x^{2} - 8$
779.2.a.b	$x^{8} + 11x^{7} - 93x^{6} - 993x^{5} + 3170x^{4} + 24514x^{3} - 44126x^{2} - 106569x - 41726$
779.2.a.c	$x^{11} + 9x^{10} - 109x^{9} - 1041x^{8} + 2758x^{7} + 29254x^{6} - 36354x^{5} - 304815x^{4} + 223266x^{3} + 1291048x^{2} - 487080x - 1802132$
779.2.a.d	$x^{19} - 15x^{18} - 289x^{17} + 4911x^{16} + 30895x^{15} - 643790x^{14} - 1455180x^{13} + 43881228x^{12} + 23652903x^{11} - 1702106023x^{10} + 374243163x^{9} + 38566787331x^{8} - 19151027884x^{7} - 495223229072x^{6} + 271842698001x^{5} + 3189763030515x^{4} - 1964722207677x^{3} - 7029599166900x^{2} + 8118789555532x - 2264036666916$
779.2.a.e	$x^{21} - 4x^{20} - 494x^{19} + 2202x^{18} + 98314x^{17} - 464203x^{16} - 10328586x^{15} + 49197482x^{14} + 637460319x^{13} - 2909802824x^{12} - 24377349180x^{11} + 99555366948x^{10} + 594505031441x^{9} - 1950764145046x^{8} - 9337021082703x^{7} + 20533323818210x^{6} + 91705599146252x^{5} - 93348268175541x^{4} - 505027930998024x^{3} - 10977444797968x^{2} + 1139733849460520x + 871734439705900$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(779))$	61
Old forms	8
Cusp forms $S_{2}(\Gamma_0(779))$	69
Eisenstein series $E_{2}(\Gamma_0(779))$	3
Modular forms $M_{2}(\Gamma_0(779))$	72

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(779)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.25 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, 19, 41, 779]: [0, 2, 4, 72] (certified at ell = 2147482801)
T = [(2, 1), (3, 3)]; h of degree 61 = 1 x 61 (phi(ord eps) x dim S_k^new); 8 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(779, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 779:2 1000

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