Atlas › Modular forms › Weight 2 › Level 487 › 487.2.a

Newspace 487.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(487))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
487.2.a.a	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{3} - 4q^{4} + 3q^{5} + O(q^{8})$	−	$+1$
487.2.a.b	2	$\mathbb{Q}(\sqrt{13})$	$2q + 3q^{3} - 4q^{4} + q^{5} - 2q^{7} + O(q^{8})$	−	$+1$
487.2.a.c	3	degree 3	$3q + 6q^{3} + 4q^{4} + 6q^{5} + 6q^{7} + O(q^{8})$	−	$+1$
487.2.a.d	16	degree 16	$16q + 7q^{2} - 4q^{3} + 27q^{4} + 9q^{5} + 2q^{6} + q^{7} + O(q^{8})$	−	$+1$
487.2.a.e	17	degree 17	$17q - 8q^{2} - 6q^{3} + 16q^{4} - 19q^{5} - 8q^{6} - 11q^{7} + O(q^{8})$	+	$-1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
487.2.a.a	$x^{2} + x - 3$
487.2.a.b	$x^{2} - 3x - 1$
487.2.a.c	$x^{3} - 6x^{2} + 7x + 5$
487.2.a.d	$x^{16} - 3x^{15} - 62x^{14} + 177x^{13} + 1385x^{12} - 3618x^{11} - 14468x^{10} + 33543x^{9} + 74563x^{8} - 149223x^{7} - 176055x^{6} + 294140x^{5} + 154217x^{4} - 174418x^{3} - 58369x^{2} + 18832x + 885$
487.2.a.e	$x^{17} + 14x^{16} + 51x^{15} - 100x^{14} - 888x^{13} - 603x^{12} + 4102x^{11} + 4468x^{10} - 8977x^{9} - 8253x^{8} + 10381x^{7} + 4364x^{6} - 4762x^{5} - 300x^{4} + 627x^{3} - 63x^{2} - 10x + 1$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(487))$	40
Old forms	0
Cusp forms $S_{2}(\Gamma_0(487))$	40
Eisenstein series $E_{2}(\Gamma_0(487))$	1
Modular forms $M_{2}(\Gamma_0(487))$	41

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.196 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, 487]: [0, 41] (certified at ell = 2147481883)
T = [(2, 1), (3, 1)]; h of degree 40 = 1 x 40 (phi(ord eps) x dim S_k^new); 4 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(487, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 487:2 1000

← 486.2.a · 488.2.a →