Atlas › Modular forms › Weight 2 › Level 482 › 482.2.a

Newspace 482.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(482))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
482.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} - q^{5} + 2q^{6} + q^{7} + O(q^{8})$	+ +	$-1$	482a1 rank 1
482.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 3q^{5} - 2q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
482.2.a.c	3	degree 3	$3q - 3q^{2} - 2q^{3} + 3q^{4} + 2q^{5} + 2q^{6} - 9q^{7} + O(q^{8})$	+ +	$-1$
482.2.a.d	6	degree 6	$6q - 6q^{2} + 2q^{3} + 6q^{4} - 5q^{5} - 2q^{6} + 10q^{7} + O(q^{8})$	+ −	$+1$
482.2.a.e	9	degree 9	$9q + 9q^{2} + 4q^{3} + 9q^{4} + 5q^{5} + 4q^{6} + 6q^{7} + O(q^{8})$	− +	$+1$

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 $T$
482.2.a.a	$x + 3$
482.2.a.b	$x^{2} + 5x + 5$
482.2.a.c	$x^{3} - 4x - 1$
482.2.a.d	$x^{6} + 3x^{5} - 34x^{4} - 65x^{3} + 222x^{2} + 451x + 97$
482.2.a.e	$x^{9} - 9x^{8} - x^{7} + 224x^{6} - 674x^{5} + 30x^{4} + 2485x^{3} - 3577x^{2} + 1599x - 122$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(482))$	21
Old forms	38
Cusp forms $S_{2}(\Gamma_0(482))$	59
Eisenstein series $E_{2}(\Gamma_0(482))$	3
Modular forms $M_{2}(\Gamma_0(482))$	62

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(482)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(241))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.292 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, 241, 482]: [0, 1, 20, 62] (certified at ell = 2147482801)
T = [(3, 1), (5, 1)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 3 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(482, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 482:2 1000

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