Atlas › Modular forms › Weight 6 › Level 95 › 95.6.a

Newspace 95.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(95))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
95.6.a.a	1	$\mathbb{Q}$	$q - 7q^{2} - 11q^{3} + 17q^{4} - 25q^{5} + 77q^{6} - 197q^{7} + O(q^{8})$	+ −	$+1$
95.6.a.b	5	degree 5	$5q - 7q^{2} - 16q^{3} + 81q^{4} + 125q^{5} - 273q^{6} - 312q^{7} + O(q^{8})$	− −	$-1$
95.6.a.c	6	degree 6	$6q + 5q^{2} - 20q^{3} + 53q^{4} - 150q^{5} + 55q^{6} - 80q^{7} + O(q^{8})$	+ +	$-1$
95.6.a.d	9	degree 9	$9q + 4q^{2} + 9q^{3} + 164q^{4} - 225q^{5} - 98q^{6} + 313q^{7} + O(q^{8})$	+ −	$+1$
95.6.a.e	9	degree 9	$9q + 9q^{2} + 38q^{3} + 209q^{4} + 225q^{5} + 91q^{6} + 80q^{7} + O(q^{8})$	− +	$+1$

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 $T$
95.6.a.a	$x + 18$
95.6.a.b	$x^{5} + 23x^{4} - 370x^{3} - 6720x^{2} + 44204x + 352232$
95.6.a.c	$x^{6} + 15x^{5} - 776x^{4} - 5420x^{3} + 134596x^{2} + 241300x - 1043304$
95.6.a.d	$x^{9} - 13x^{8} - 1808x^{7} + 14560x^{6} + 1086375x^{5} - 3082409x^{4} - 238100942x^{3} - 561317760x^{2} + 11448718324x + 45528268372$
95.6.a.e	$x^{9} - 47x^{8} - 852x^{7} + 56656x^{6} + 127157x^{5} - 22660641x^{4} + 31500410x^{3} + 3401007992x^{2} - 4654753356x - 148052592456$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(95))$	30
Old forms	18
Cusp forms $S_{6}(\Gamma_0(95))$	48
Eisenstein series $E_{6}(\Gamma_0(95))$	4
Modular forms $M_{6}(\Gamma_0(95))$	52

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(95)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.266 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, 5, 19, 95]: [1, 3, 10, 52] (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(95, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 95:6 1000

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