Atlas › Modular forms › Weight 4 › Level 93 › 93.4.a

Newspace 93.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(93))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
93.4.a.a	1	$\mathbb{Q}$	$q + 3q^{2} - 3q^{3} + q^{4} - 9q^{5} - 9q^{6} - 34q^{7} + O(q^{8})$	+ −	$-1$
93.4.a.b	2	$\mathbb{Q}(\sqrt{29})$	$2q - 5q^{2} - 6q^{3} + 11q^{4} + 8q^{5} + 15q^{6} - 4q^{7} + O(q^{8})$	+ −	$-1$
93.4.a.c	2	$\mathbb{Q}(\sqrt{17})$	$2q - 3q^{2} + 6q^{3} - 3q^{4} - 8q^{5} - 9q^{6} - 37q^{7} + O(q^{8})$	− +	$-1$
93.4.a.d	2	$\mathbb{Q}(\sqrt{41})$	$2q - q^{2} - 6q^{3} + 5q^{4} - 11q^{5} + 3q^{6} + 29q^{7} + O(q^{8})$	+ −	$-1$
93.4.a.e	3	degree 3	$3q + 3q^{2} - 9q^{3} + 5q^{4} + 8q^{5} - 9q^{6} + 19q^{7} + O(q^{8})$	+ +	$+1$
93.4.a.f	6	degree 6	$6q + 3q^{2} + 18q^{3} + 33q^{4} + 12q^{5} + 9q^{6} + 47q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
93.4.a.a	$x + 6$
93.4.a.b	$x^{2} - 3x - 353$
93.4.a.c	$x^{2} + 11x + 26$
93.4.a.d	$x^{2} + 12x - 5$
93.4.a.e	$x^{3} - 11x^{2} - 112x + 964$
93.4.a.f	$x^{6} - 15x^{5} - 370x^{4} + 7611x^{3} - 27923x^{2} - 57444x - 17796$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(93))$	16
Old forms	14
Cusp forms $S_{4}(\Gamma_0(93))$	30
Eisenstein series $E_{4}(\Gamma_0(93))$	4
Modular forms $M_{4}(\Gamma_0(93))$	34

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(93)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.108 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, 31, 93]: [1, 2, 9, 34] (certified at ell = 2147482951)
T = [(2, 1), (5, 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(93, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 93:4 1000

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