Atlas › Modular forms › Weight 6 › Level 77 › 77.6.a

Newspace 77.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(77))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
77.6.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 6q^{3} - 28q^{4} - 74q^{5} + 12q^{6} - 49q^{7} + O(q^{8})$	+ −	$+1$
77.6.a.b	4	degree 4	$4q - 8q^{2} - 43q^{3} - 10q^{4} + 57q^{5} + 166q^{6} + 196q^{7} + O(q^{8})$	− −	$-1$
77.6.a.c	5	degree 5	$5q + 10q^{2} + 17q^{3} + 114q^{4} + 119q^{5} - 112q^{6} - 245q^{7} + O(q^{8})$	+ −	$+1$
77.6.a.d	6	degree 6	$6q + 4q^{2} - 12q^{3} + 70q^{4} - 62q^{5} - 172q^{6} - 294q^{7} + O(q^{8})$	+ +	$-1$
77.6.a.e	8	degree 8	$8q + 4q^{2} + 42q^{3} + 166q^{4} + 50q^{5} + 238q^{6} + 392q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
77.6.a.a	$x + 8$
77.6.a.b	$x^{4} + 51x^{3} + 569x^{2} - 975x - 4454$
77.6.a.c	$x^{5} - 27x^{4} - 735x^{3} + 14031x^{2} + 151998x - 56628$
77.6.a.d	$x^{6} + 8x^{5} - 782x^{4} + 1790x^{3} + 74177x^{2} - 173142x - 602052$
77.6.a.e	$x^{8} - 46x^{7} - 632x^{6} + 47440x^{5} - 126953x^{4} - 13089262x^{3} + 98203060x^{2} + 444736128x - 1190863872$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(77))$	24
Old forms	14
Cusp forms $S_{6}(\Gamma_0(77))$	38
Eisenstein series $E_{6}(\Gamma_0(77))$	4
Modular forms $M_{6}(\Gamma_0(77))$	42

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(77)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.251 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, 7, 11, 77]: [1, 5, 6, 42] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; h of degree 24 = 1 x 24 (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(77, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 77:6 1000

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