Atlas › Modular forms › Weight 2 › Level 287 › 287.2.a

Newspace 287.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(287))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
287.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - 3q^{3} - q^{4} - q^{5} + 4q^{6} + 2q^{7} + O(q^{8})$	− −	$-1$
287.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - q^{3} - q^{4} + q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
287.2.a.c	3	degree 3	$3q + q^{2} - q^{3} + 3q^{4} + 6q^{5} - 8q^{6} + 3q^{7} + O(q^{8})$	− +	$+1$
287.2.a.d	3	degree 3	$3q + 4q^{2} + 5q^{3} + 4q^{4} + 2q^{5} + 2q^{6} - 3q^{7} + O(q^{8})$	+ −	$+1$
287.2.a.e	5	degree 5	$5q - q^{2} + 4q^{3} + 3q^{4} - 5q^{5} + 12q^{6} + 5q^{7} + O(q^{8})$	− +	$+1$
287.2.a.f	6	degree 6	$6q - q^{2} - 4q^{3} + 9q^{4} - q^{5} - 4q^{6} - 6q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
287.2.a.a	$x^{2} + 10x + 5$
287.2.a.b	$x^{2} + 4x - 1$
287.2.a.c	$x^{3} + 2x^{2} - 55x + 57$
287.2.a.d	$x^{3} - 19x^{2} + 111x - 197$
287.2.a.e	$x^{5} - 11x^{4} - 54x^{3} + 554x^{2} + 453x - 1167$
287.2.a.f	$x^{6} + 13x^{5} - 58x^{4} - 1219x^{3} - 1360x^{2} + 27138x + 78485$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(287))$	21
Old forms	6
Cusp forms $S_{2}(\Gamma_0(287))$	27
Eisenstein series $E_{2}(\Gamma_0(287))$	3
Modular forms $M_{2}(\Gamma_0(287))$	30

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(287)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.177 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, 7, 41, 287]: [0, 1, 4, 30] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; 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(287, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 287:2 1000

← 286.2.a · 288.2.a →