Atlas › Modular forms › Weight 2 › Level 515 › 515.2.a

Newspace 515.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(515))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
515.2.a.a	4	degree 4	$4q - q^{2} - q^{3} - q^{4} + 4q^{5} - 4q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$
515.2.a.b	8	degree 8	$8q - 2q^{2} - 3q^{3} + 6q^{4} - 8q^{5} - 8q^{6} - 11q^{7} + O(q^{8})$	+ +	$-1$
515.2.a.c	9	degree 9	$9q + q^{2} + 3q^{3} + 11q^{4} - 9q^{5} + 9q^{7} + O(q^{8})$	+ −	$+1$
515.2.a.d	14	degree 14	$14q - q^{2} + q^{3} + 17q^{4} + 14q^{5} + 4q^{6} + 9q^{7} + O(q^{8})$	− +	$+1$

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 $T$
515.2.a.a	$x^{4} + 4x^{3} - 15x^{2} - 28x - 11$
515.2.a.b	$x^{8} + 11x^{7} - 84x^{6} - 898x^{5} + 2725x^{4} + 19830x^{3} - 45660x^{2} - 67401x + 103797$
515.2.a.c	$x^{9} - 10x^{8} - 104x^{7} + 1017x^{6} + 3802x^{5} - 31389x^{4} - 54431x^{3} + 260785x^{2} + 89626x - 313002$
515.2.a.d	$x^{14} - 2x^{13} - 334x^{12} + 969x^{11} + 41378x^{10} - 151260x^{9} - 2302339x^{8} + 9867009x^{7} + 54522929x^{6} - 267953528x^{5} - 400200865x^{4} + 2658137873x^{3} - 809698201x^{2} - 6477808198x + 5834971822$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(515))$	35
Old forms	16
Cusp forms $S_{2}(\Gamma_0(515))$	51
Eisenstein series $E_{2}(\Gamma_0(515))$	3
Modular forms $M_{2}(\Gamma_0(515))$	54

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(515)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(103))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.205 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, 103, 515]: [0, 1, 9, 54] (certified at ell = 2147482501)
T = [(2, 1), (3, 3)]; h of degree 35 = 1 x 35 (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(515, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 515:2 1000

← 514.2.a · 516.2.a →