Atlas › Modular forms › Weight 6 › Level 82 › 82.6.a

Newspace 82.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(82))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
82.6.a.a	3	degree 3	$3q + 12q^{2} - 28q^{3} + 48q^{4} - 26q^{5} - 112q^{6} - 158q^{7} + O(q^{8})$	− −	$-1$
82.6.a.b	4	degree 4	$4q - 16q^{2} + 8q^{3} + 64q^{4} + 12q^{5} - 32q^{6} + 158q^{7} + O(q^{8})$	+ −	$+1$
82.6.a.c	5	degree 5	$5q - 20q^{2} - 10q^{3} + 80q^{4} - 38q^{5} + 40q^{6} - 38q^{7} + O(q^{8})$	+ +	$-1$
82.6.a.d	6	degree 6	$6q + 24q^{2} + 8q^{3} + 96q^{4} + 124q^{5} + 32q^{6} + 38q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
82.6.a.a	$x^{3} + 54x^{2} + 612x + 1018$
82.6.a.b	$x^{4} - 20x^{3} - 11792x^{2} + 217486x + 5613840$
82.6.a.c	$x^{5} + 48x^{4} - 2492x^{3} - 117954x^{2} - 737450x + 8103036$
82.6.a.d	$x^{6} - 132x^{5} - 9022x^{4} + 1740714x^{3} - 54907538x^{2} - 110043672x + 611151084$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(82))$	18
Old forms	32
Cusp forms $S_{6}(\Gamma_0(82))$	50
Eisenstein series $E_{6}(\Gamma_0(82))$	4
Modular forms $M_{6}(\Gamma_0(82))$	54

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(82)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.256 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, 2, 41, 82]: [1, 2, 18, 54] (certified at ell = 2147482921)
T = [(3, 1), (5, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 5 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(82, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 82:6 1000

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