Atlas › Modular forms › Weight 4 › Level 77 › 77.4.a

Newspace 77.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(77))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
77.4.a.a	1	$\mathbb{Q}$	$q + 3q^{2} + 4q^{3} + q^{4} + 12q^{5} + 12q^{6} + 7q^{7} + O(q^{8})$	− −	$+1$
77.4.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 4q^{3} + 2q^{4} - 4q^{5} + 4q^{6} + 14q^{7} + O(q^{8})$	− +	$-1$
77.4.a.c	4	degree 4	$4q - 4q^{2} - 12q^{3} + 22q^{4} - 18q^{5} + 2q^{6} - 28q^{7} + O(q^{8})$	+ −	$-1$
77.4.a.d	4	degree 4	$4q - 2q^{2} + 14q^{3} + 26q^{4} + 10q^{5} + 14q^{6} - 28q^{7} + O(q^{8})$	+ +	$+1$
77.4.a.e	5	degree 5	$5q + q^{2} + 2q^{3} + 45q^{4} - 24q^{5} + 4q^{6} + 35q^{7} + O(q^{8})$	− −	$+1$

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 $T$
77.4.a.a	$x - 7$
77.4.a.b	$x^{2} + 6x + 1$
77.4.a.c	$x^{4} + 16x^{3} + 12x^{2} - 100x + 63$
77.4.a.d	$x^{4} - 12x^{3} - 63x^{2} + 888x - 1266$
77.4.a.e	$x^{5} - 3x^{4} - 141x^{3} + 23x^{2} + 2634x - 1002$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(77))$	16
Old forms	6
Cusp forms $S_{4}(\Gamma_0(77))$	22
Eisenstein series $E_{4}(\Gamma_0(77))$	4
Modular forms $M_{4}(\Gamma_0(77))$	26

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(77)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.104 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, 3, 4, 26] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; 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(77, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 77:4 1000

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