Atlas › Modular forms › Weight 4 › Level 82 › 82.4.a

Newspace 82.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(82))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
82.4.a.a	1	$\mathbb{Q}$	$q + 2q^{2} - 4q^{3} + 4q^{4} - 18q^{5} - 8q^{6} - 2q^{7} + O(q^{8})$	− +	$-1$
82.4.a.b	1	$\mathbb{Q}$	$q + 2q^{2} + 10q^{3} + 4q^{4} - 6q^{5} + 20q^{6} - 10q^{7} + O(q^{8})$	− −	$+1$
82.4.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q - 4q^{2} + 2q^{3} + 8q^{4} + 8q^{5} - 4q^{6} + 2q^{7} + O(q^{8})$	+ +	$+1$
82.4.a.d	3	degree 3	$3q - 6q^{2} - 4q^{3} + 12q^{4} - 2q^{5} + 8q^{6} - 26q^{7} + O(q^{8})$	+ −	$-1$
82.4.a.e	3	degree 3	$3q + 6q^{2} - 2q^{3} + 12q^{4} + 18q^{5} - 4q^{6} + 36q^{7} + O(q^{8})$	− −	$+1$

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 $T$
82.4.a.a	$x + 22$
82.4.a.b	$x - 4$
82.4.a.c	$x^{2} - 10x - 122$
82.4.a.d	$x^{3} + 6x^{2} - 200x - 1050$
82.4.a.e	$x^{3} - 16x^{2} - 20x - 6$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(82))$	10
Old forms	20
Cusp forms $S_{4}(\Gamma_0(82))$	30
Eisenstein series $E_{4}(\Gamma_0(82))$	4
Modular forms $M_{4}(\Gamma_0(82))$	34

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(82)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.105 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, 12, 34] (certified at ell = 2147482921)
T = [(3, 1), (5, 1)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 2 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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 82:4 1000

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