Atlas › Modular forms › Weight 4 › Level 117 › 117.4.a

Newspace 117.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(117))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
117.4.a.a	1	$\mathbb{Q}$	$q - 8q^{4} + 12q^{5} + 2q^{7} + O(q^{8})$	−
117.4.a.b	1	$\mathbb{Q}$	$q + 5q^{2} + 17q^{4} + 7q^{5} - 13q^{7} + O(q^{8})$	−
117.4.a.c	2	$\mathbb{Q}(\sqrt{14})$	$2q - 2q^{2} + 14q^{4} - 24q^{5} + O(q^{8})$	+
117.4.a.d	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{2} - 7q^{4} + 3q^{5} - 9q^{7} + O(q^{8})$	+
117.4.a.e	2	$\mathbb{Q}(\sqrt{7})$	$2q - 2q^{4} - 44q^{7} + O(q^{8})$	−
117.4.a.f	3	degree 3	$3q - 2q^{2} + 10q^{4} - 4q^{5} + 30q^{7} + O(q^{8})$	−
117.4.a.g	4	degree 4	$4q + 58q^{4} + 36q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
117.4.a.a	$x - 36$
117.4.a.b	$x - 26$
117.4.a.c	$x^{2} + 74x + 1019$
117.4.a.d	$x^{2} - 8x - 52$
117.4.a.e	$x^{2} - 847$
117.4.a.f	$x^{3} + 14x^{2} - 2247x + 27696$
117.4.a.g	$x^{4} - 5253x^{2} + 5273968$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(117))$	15
Old forms	23
Cusp forms $S_{4}(\Gamma_0(117))$	38
Eisenstein series $E_{4}(\Gamma_0(117))$	8
Modular forms $M_{4}(\Gamma_0(117))$	46

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(117)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.118 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, 3, 9, 13, 39, 117]: [1, 2, 4, 5, 16, 44] (certified at ell = 2147483629)
T = [(2, 1), (5, 3)]; h of degree 15 = 1 x 15 (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(117, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 117:4 1000

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