Atlas › Modular forms › Weight 6 › Level 56 › 56.6.a

Newspace 56.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(56))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
56.6.a.a	1	$\mathbb{Q}$	$q - 6q^{3} + 4q^{5} + 49q^{7} + O(q^{8})$	−
56.6.a.b	1	$\mathbb{Q}$	$q + 30q^{3} + 32q^{5} + 49q^{7} + O(q^{8})$	−
56.6.a.c	2	$\mathbb{Q}(\sqrt{177})$	$2q - 26q^{3} - 62q^{5} + 98q^{7} + O(q^{8})$	−
56.6.a.d	2	$\mathbb{Q}(\sqrt{193})$	$2q - 14q^{3} + 42q^{5} - 98q^{7} + O(q^{8})$	+
56.6.a.e	2	$\mathbb{Q}(\sqrt{345})$	$2q - 6q^{3} + 82q^{5} - 98q^{7} + O(q^{8})$	+

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 $T$
56.6.a.a	$x + 2$
56.6.a.b	$x - 62$
56.6.a.c	$x^{2} + 88x - 4436$
56.6.a.d	$x^{2} - 28x - 2892$
56.6.a.e	$x^{2} - 76x - 4076$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(56))$	8
Old forms	28
Cusp forms $S_{6}(\Gamma_0(56))$	36
Eisenstein series $E_{6}(\Gamma_0(56))$	8
Modular forms $M_{6}(\Gamma_0(56))$	44

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(56)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(4))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 4}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 3}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(28))^{\oplus 2}$

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, 2, 4, 7, 8, 14, 28, 56]: [1, 2, 4, 5, 7, 12, 23, 44] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 8 = 1 x 8 (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(56, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 56:6 1000

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