Atlas › Modular forms › Weight 8 › Level 56 › 56.8.a

Newspace 56.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(56))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
56.8.a.a	1	$\mathbb{Q}$	$q - 18q^{3} + 160q^{5} - 343q^{7} + O(q^{8})$	+
56.8.a.b	1	$\mathbb{Q}$	$q + 46q^{3} - 160q^{5} - 343q^{7} + O(q^{8})$	+
56.8.a.c	2	$\mathbb{Q}(\sqrt{249})$	$2q - 42q^{3} + 14q^{5} + 686q^{7} + O(q^{8})$	−
56.8.a.d	3	degree 3	$3q + 12q^{3} - 598q^{5} - 1029q^{7} + O(q^{8})$	+
56.8.a.e	3	degree 3	$3q + 28q^{3} + 138q^{5} + 1029q^{7} + O(q^{8})$	−

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 $T$
56.8.a.a	$x - 142$
56.8.a.b	$x + 114$
56.8.a.c	$x^{2} + 28x - 15740$
56.8.a.d	$x^{3} + 586x^{2} + 38300x - 10484200$
56.8.a.e	$x^{3} - 166x^{2} - 214708x + 46620664$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(56))$	10
Old forms	42
Cusp forms $S_{8}(\Gamma_0(56))$	52
Eisenstein series $E_{8}(\Gamma_0(56))$	8
Modular forms $M_{8}(\Gamma_0(56))$	60

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.313 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, 3, 5, 5, 9, 16, 31, 60] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 4 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, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 56:8 1000

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