Atlas › Modular forms › Weight 4 › Level 91 › 91.4.a

Newspace 91.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(91))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
91.4.a.a	3	degree 3	$3q + q^{2} + q^{3} - 3q^{4} - 22q^{5} - 23q^{6} - 21q^{7} + O(q^{8})$	+ −	$-1$
91.4.a.b	4	degree 4	$4q - 4q^{2} - 5q^{3} + 26q^{4} - 36q^{5} - 45q^{6} + 28q^{7} + O(q^{8})$	− +	$-1$
91.4.a.c	5	degree 5	$5q + 7q^{2} - 5q^{3} + 53q^{4} + 16q^{5} + q^{6} - 35q^{7} + O(q^{8})$	+ +	$+1$
91.4.a.d	6	degree 6	$6q + 2q^{2} + 13q^{3} + 10q^{4} + 26q^{5} + 3q^{6} + 42q^{7} + O(q^{8})$	− −	$+1$

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 $T$
91.4.a.a	$x^{3} - 2x^{2} - 10x + 4$
91.4.a.b	$x^{4} + 9x^{3} - 8x^{2} - 66x + 72$
91.4.a.c	$x^{5} - 2x^{4} - 142x^{3} + 36x^{2} + 4844x + 11304$
91.4.a.d	$x^{6} - 15x^{5} - 56x^{4} + 1342x^{3} - 2656x^{2} - 16508x + 47232$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(91))$	18
Old forms	8
Cusp forms $S_{4}(\Gamma_0(91))$	26
Eisenstein series $E_{4}(\Gamma_0(91))$	4
Modular forms $M_{4}(\Gamma_0(91))$	30

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(91)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 2}$

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, 13, 91]: [1, 3, 5, 30] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 18 = 1 x 18 (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(91, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 91:4 1000

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