Atlas › Modular forms › Weight 2 › Level 56 › 56.2.a

Newspace 56.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(56))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
56.2.a.a	1	$\mathbb{Q}$	$q + 2q^{5} - q^{7} + O(q^{8})$	+			56a1 rank 0
56.2.a.b	1	$\mathbb{Q}$	$q + 2q^{3} - 4q^{5} + q^{7} + O(q^{8})$	−			56b1 rank 0

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 $T$
56.2.a.a	$x - 6$
56.2.a.b	$x + 10$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(56))$	2
Old forms	3
Cusp forms $S_{2}(\Gamma_0(56))$	5
Eisenstein series $E_{2}(\Gamma_0(56))$	7
Modular forms $M_{2}(\Gamma_0(56))$	12

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(56)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 3}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.088 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, 7, 8, 14, 28, 56]: [0, 1, 2, 1, 3, 4, 7, 12] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 2 = 1 x 2 (phi(ord eps) x dim S_k^new); 1 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(56, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 56:2 1000

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