Atlas › Modular forms › Weight 2 › Level 455 › 455.2.a

Newspace 455.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(455))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
455.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + q^{5} - q^{7} + O(q^{8})$	− + −	$-1$	455b1 rank 1
455.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - q^{5} - q^{7} + O(q^{8})$	+ + +	$-1$	455a1 rank 1
455.2.a.c	4	degree 4	$4q - q^{2} + 3q^{4} - 4q^{5} + 6q^{6} - 4q^{7} + O(q^{8})$	+ + −	$+1$
455.2.a.d	4	degree 4	$4q + 3q^{2} + 4q^{3} + 3q^{4} + 4q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	− + +	$+1$
455.2.a.e	6	degree 6	$6q + 3q^{2} + 9q^{4} + 6q^{5} - 6q^{6} + 6q^{7} + O(q^{8})$	− − −	$+1$
455.2.a.f	7	degree 7	$7q + 16q^{4} - 7q^{5} + 2q^{6} + 7q^{7} + O(q^{8})$	+ − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
455.2.a.a	$x + 1$
455.2.a.b	$x - 1$
455.2.a.c	$x^{4} + x^{3} - 104x^{2} - 162x + 1567$
455.2.a.d	$x^{4} - 15x^{3} + 20x^{2} + 512x - 1811$
455.2.a.e	$x^{6} - 3x^{5} - 105x^{4} + 179x^{3} + 2601x^{2} - 4006x - 10311$
455.2.a.f	$x^{7} - 210x^{5} + 206x^{4} + 11190x^{3} - 12791x^{2} - 110793x + 39661$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(455))$	23
Old forms	30
Cusp forms $S_{2}(\Gamma_0(455))$	53
Eisenstein series $E_{2}(\Gamma_0(455))$	7
Modular forms $M_{2}(\Gamma_0(455))$	60

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(455)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(65))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(91))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.316 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, 7, 13, 35, 65, 91, 455]: [0, 1, 1, 1, 6, 8, 10, 60] (certified at ell = 2147483629)
T = [(2, 1), (3, 3)]; h of degree 23 = 1 x 23 (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(455, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 455:2 1000

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