Atlas › Modular forms › Weight 4 › Level 178 › 178.4.a

Newspace 178.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(178))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
178.4.a.a	3	degree 3	$3q + 6q^{2} - 9q^{3} + 12q^{4} - 19q^{5} - 18q^{6} - 2q^{7} + O(q^{8})$	− +	$-1$
178.4.a.b	5	degree 5	$5q - 10q^{2} + 3q^{3} + 20q^{4} + 17q^{5} - 6q^{6} + 2q^{7} + O(q^{8})$	+ +	$+1$
178.4.a.c	6	degree 6	$6q - 12q^{2} - 9q^{3} + 24q^{4} - 3q^{5} + 18q^{6} - 26q^{7} + O(q^{8})$	+ −	$-1$
178.4.a.d	8	degree 8	$8q + 16q^{2} + 9q^{3} + 32q^{4} + 21q^{5} + 18q^{6} + 26q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
178.4.a.a	$x^{3} + 28x^{2} + 214x + 246$
178.4.a.b	$x^{5} - 20x^{4} - 448x^{3} + 7970x^{2} + 11600x - 12000$
178.4.a.c	$x^{6} + 12x^{5} - 254x^{4} - 2582x^{3} + 12648x^{2} + 110862x - 74628$
178.4.a.d	$x^{8} - 30x^{7} - 450x^{6} + 20250x^{5} - 112192x^{4} - 1098614x^{3} + 8492192x^{2} - 9267600x - 15537600$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(178))$	22
Old forms	44
Cusp forms $S_{4}(\Gamma_0(178))$	66
Eisenstein series $E_{4}(\Gamma_0(178))$	4
Modular forms $M_{4}(\Gamma_0(178))$	70

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(178)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(89))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.126 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, 89, 178]: [1, 2, 24, 70] (certified at ell = 2147482921)
T = [(3, 1), (5, 1)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 4 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(178, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 178:4 1000

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