Atlas › Modular forms › Weight 2 › Level 808 › 808.2.a

Newspace 808.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(808))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
808.2.a.a	1	$\mathbb{Q}$	$q - 2q^{5} - q^{7} + O(q^{8})$	+	808a1 rank 0
808.2.a.b	1	$\mathbb{Q}$	$q + 2q^{3} + 3q^{5} + 2q^{7} + O(q^{8})$	+	808b1 rank 0
808.2.a.c	3	degree 3	$3q - q^{3} + q^{5} + 5q^{7} + O(q^{8})$	+
808.2.a.d	4	degree 4	$4q - q^{3} - 4q^{5} + q^{7} + O(q^{8})$	+
808.2.a.e	7	degree 7	$7q - 2q^{3} - 3q^{5} - 10q^{7} + O(q^{8})$	−
808.2.a.f	9	degree 9	$9q + 2q^{3} + 5q^{5} - q^{7} + O(q^{8})$	−

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 $T$
808.2.a.a	$x + 2$
808.2.a.b	$x - 5$
808.2.a.c	$x^{3} - 19x + 19$
808.2.a.d	$x^{4} + 5x^{3} + x^{2} - 4x + 1$
808.2.a.e	$x^{7} + 5x^{6} - 4x^{5} - 44x^{4} - 39x^{3} + 9x^{2} + 9x + 1$
808.2.a.f	$x^{9} - 7x^{8} - 24x^{7} + 236x^{6} - 203x^{5} - 835x^{4} + 1061x^{3} - 147x^{2} - 68x - 4$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(808))$	25
Old forms	74
Cusp forms $S_{2}(\Gamma_0(808))$	99
Eisenstein series $E_{2}(\Gamma_0(808))$	7
Modular forms $M_{2}(\Gamma_0(808))$	106

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(808)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(101))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(202))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(404))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.253 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, 4, 8, 101, 202, 404, 808]: [0, 1, 2, 3, 9, 27, 54, 106] (certified at ell = 2147482801)
T = [(3, 1), (5, 1)]; h of degree 25 = 1 x 25 (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(808, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 808:2 1000

← 807.2.a · 809.2.a →