Atlas › Modular forms › Weight 2 › Level 556 › 556.2.a

Newspace 556.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(556))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
556.2.a.a	1	$\mathbb{Q}$	$q - q^{5} - q^{7} + O(q^{8})$	−	556a1 rank 1
556.2.a.b	3	degree 3	$3q - 2q^{3} - 4q^{7} + O(q^{8})$	−
556.2.a.c	7	degree 7	$7q + 4q^{3} - q^{5} + q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
556.2.a.a	$x + 3$
556.2.a.b	$x^{3} + 2x^{2} - 63x - 147$
556.2.a.c	$x^{7} - x^{6} - 264x^{5} + 622x^{4} + 15389x^{3} - 32067x^{2} - 106419x + 210667$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(556))$	11
Old forms	57
Cusp forms $S_{2}(\Gamma_0(556))$	68
Eisenstein series $E_{2}(\Gamma_0(556))$	5
Modular forms $M_{2}(\Gamma_0(556))$	73

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(556)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(139))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(278))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.208 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, 139, 278, 556]: [0, 1, 2, 12, 37, 73] (certified at ell = 2147483137)
T = [(3, 1), (5, 3)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 2 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(556, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 556:2 1000

← 555.2.a · 557.2.a →