Atlas › Modular forms › Weight 4 › Level 158 › 158.4.a

Newspace 158.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(158))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
158.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 3q^{3} + 4q^{4} - 9q^{5} - 6q^{6} + 9q^{7} + O(q^{8})$	+ −	$-1$
158.4.a.b	3	degree 3	$3q - 6q^{2} - 3q^{3} + 12q^{4} + 6q^{5} + 6q^{6} - 39q^{7} + O(q^{8})$	+ −	$-1$
158.4.a.c	4	degree 4	$4q + 8q^{2} - 12q^{3} + 16q^{4} - 17q^{5} - 24q^{6} - 58q^{7} + O(q^{8})$	− +	$-1$
158.4.a.d	6	degree 6	$6q - 12q^{2} + 6q^{3} + 24q^{4} - 3q^{5} - 12q^{6} + 40q^{7} + O(q^{8})$	+ +	$+1$
158.4.a.e	6	degree 6	$6q + 12q^{2} + 12q^{3} + 24q^{4} + 23q^{5} + 24q^{6} + 40q^{7} + O(q^{8})$	− −	$+1$

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 $T$
158.4.a.a	$x + 6$
158.4.a.b	$x^{3} - 3x^{2} - 73x + 278$
158.4.a.c	$x^{4} + 29x^{3} + 145x^{2} - 1458x - 7996$
158.4.a.d	$x^{6} - 3x^{5} - 691x^{4} + 4522x^{3} + 97596x^{2} - 816480x - 23328$
158.4.a.e	$x^{6} - 35x^{5} + 49x^{4} + 8352x^{3} - 55918x^{2} - 458596x + 3840584$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(158))$	20
Old forms	38
Cusp forms $S_{4}(\Gamma_0(158))$	58
Eisenstein series $E_{4}(\Gamma_0(158))$	4
Modular forms $M_{4}(\Gamma_0(158))$	62

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(158)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(79))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.122 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, 79, 158]: [1, 2, 21, 62] (certified at ell = 2147483053)
T = [(3, 1), (5, 1)]; h of degree 20 = 1 x 20 (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(158, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 158:4 1000

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