Atlas › Modular forms › Weight 2 › Level 813 › 813.2.a

Newspace 813.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(813))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
813.2.a.a	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - q^{7} + O(q^{8})$	− −	$-1$	813b1 rank 1
813.2.a.b	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 2q^{4} + 2q^{5} + 2q^{6} - q^{7} + O(q^{8})$	− +	$+1$	813a1 rank 0
813.2.a.c	4	degree 4	$4q - q^{2} + 4q^{3} + q^{4} - 4q^{5} - q^{6} - 8q^{7} + O(q^{8})$	− −	$-1$
813.2.a.d	7	degree 7	$7q + q^{2} - 7q^{3} + 5q^{4} + 6q^{5} - q^{6} + 7q^{7} + O(q^{8})$	+ −	$+1$
813.2.a.e	15	degree 15	$15q - 4q^{2} - 15q^{3} + 18q^{4} - 10q^{5} + 4q^{6} - 9q^{7} + O(q^{8})$	+ +	$-1$
813.2.a.f	17	degree 17	$17q - q^{2} + 17q^{3} + 21q^{4} + 4q^{5} - q^{6} + 16q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
813.2.a.a	$x$
813.2.a.b	$x - 4$
813.2.a.c	$x^{4} + 5x^{3} + 4x^{2} - 8x - 9$
813.2.a.d	$x^{7} - 7x^{6} + 3x^{5} + 37x^{4} - x^{3} - 36x^{2} - 3x + 4$
813.2.a.e	$x^{15} + 14x^{14} + 27x^{13} - 386x^{12} - 1490x^{11} + 3434x^{10} + 18807x^{9} - 11895x^{8} - 102285x^{7} + 9096x^{6} + 270146x^{5} + 28853x^{4} - 332788x^{3} - 36678x^{2} + 145279x - 15992$
813.2.a.f	$x^{17} - 3x^{16} - 84x^{15} + 292x^{14} + 2535x^{13} - 9977x^{12} - 35387x^{11} + 163348x^{10} + 226963x^{9} - 1404139x^{8} - 382662x^{7} + 6264450x^{6} - 2316327x^{5} - 12721183x^{4} + 9268740x^{3} + 7532359x^{2} - 5041565x - 1854497$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(813))$	45
Old forms	44
Cusp forms $S_{2}(\Gamma_0(813))$	89
Eisenstein series $E_{2}(\Gamma_0(813))$	3
Modular forms $M_{2}(\Gamma_0(813))$	92

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(813)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(271))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.246 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, 3, 271, 813]: [0, 1, 23, 92] (certified at ell = 2147482801)
T = [(2, 1), (5, 1)]; h of degree 45 = 1 x 45 (phi(ord eps) x dim S_k^new); 5 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(813, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 813:2 1000

← 812.2.a · 814.2.a →