Atlas › Modular forms › Weight 2 › Level 382 › 382.2.a

Newspace 382.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(382))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
382.2.a.a	3	degree 3	$3q - 3q^{2} - q^{3} + 3q^{4} - 4q^{5} + q^{6} - q^{7} + O(q^{8})$	+ +	$-1$
382.2.a.b	3	degree 3	$3q + 3q^{2} - 5q^{3} + 3q^{4} - 6q^{5} - 5q^{6} - 5q^{7} + O(q^{8})$	− −	$-1$
382.2.a.c	4	degree 4	$4q + 4q^{2} + 3q^{3} + 4q^{4} + 4q^{5} + 3q^{6} + q^{7} + O(q^{8})$	− +	$+1$
382.2.a.d	5	degree 5	$5q - 5q^{2} + 3q^{3} + 5q^{4} + 8q^{5} - 3q^{6} + q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
382.2.a.a	$x^{3} + 5x^{2} + 4x - 5$
382.2.a.b	$x^{3} + 11x^{2} + 38x + 41$
382.2.a.c	$x^{4} - 7x^{3} + 12x^{2} + x - 2$
382.2.a.d	$x^{5} - 11x^{4} + 28x^{3} + 27x^{2} - 144x + 108$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(382))$	15
Old forms	32
Cusp forms $S_{2}(\Gamma_0(382))$	47
Eisenstein series $E_{2}(\Gamma_0(382))$	3
Modular forms $M_{2}(\Gamma_0(382))$	50

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(382)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(191))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.189 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, 191, 382]: [0, 1, 17, 50] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (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(382, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 382:2 1000

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