Atlas › Modular forms › Weight 6 › Level 91 › 91.6.a

Newspace 91.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(91))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
91.6.a.a	6	degree 6	$6q - 13q^{2} - 26q^{3} + 49q^{4} - 130q^{5} + 147q^{6} + 294q^{7} + O(q^{8})$	− −	$-1$
91.6.a.b	7	degree 7	$7q - 2q^{2} + 28q^{3} + 102q^{4} - 62q^{5} - 191q^{6} - 343q^{7} + O(q^{8})$	+ +	$-1$
91.6.a.c	8	degree 8	$8q - q^{2} + 28q^{3} + 245q^{4} + 219q^{5} + 435q^{6} + 392q^{7} + O(q^{8})$	− +	$+1$
91.6.a.d	9	degree 9	$9q + 10q^{2} + 10q^{3} + 66q^{4} + 89q^{5} - 47q^{6} - 441q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
91.6.a.a	$x^{6} + 39x^{5} - 296x^{4} - 12848x^{3} + 40016x^{2} + 602452x + 984696$
91.6.a.b	$x^{7} - 26x^{6} - 629x^{5} + 20738x^{4} - 138084x^{3} - 500004x^{2} + 8195044x - 22478352$
91.6.a.c	$x^{8} - 27x^{7} - 1758x^{6} + 50718x^{5} + 559812x^{4} - 17131764x^{3} - 53952320x^{2} + 1441170768x - 1529786880$
91.6.a.d	$x^{9} - 20x^{8} - 1573x^{7} + 22714x^{6} + 889358x^{5} - 7575628x^{4} - 204924468x^{3} + 717167488x^{2} + 16308058000x + 7436000000$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(91))$	30
Old forms	16
Cusp forms $S_{6}(\Gamma_0(91))$	46
Eisenstein series $E_{6}(\Gamma_0(91))$	4
Modular forms $M_{6}(\Gamma_0(91))$	50

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(91)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.257 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, 5, 7, 50] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 6 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, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 91:6 1000

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