Atlas › Modular forms › Weight 6 › Level 58 › 58.6.a

Newspace 58.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(58))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
58.6.a.a	2	$\mathbb{Q}(\sqrt{22})$	$2q + 8q^{2} - 14q^{3} + 32q^{4} - 82q^{5} - 56q^{6} - 160q^{7} + O(q^{8})$	− −	$-1$
58.6.a.b	3	degree 3	$3q - 12q^{2} + 14q^{3} + 48q^{4} + 68q^{5} - 56q^{6} + 24q^{7} + O(q^{8})$	+ −	$+1$
58.6.a.c	4	degree 4	$4q - 16q^{2} - 13q^{3} + 64q^{4} - 7q^{5} + 52q^{6} - 74q^{7} + O(q^{8})$	+ +	$-1$
58.6.a.d	4	degree 4	$4q + 16q^{2} + 13q^{3} + 64q^{4} + 93q^{5} + 52q^{6} + 134q^{7} + O(q^{8})$	− +	$+1$

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 $T$
58.6.a.a	$x^{2} + 96x + 522$
58.6.a.b	$x^{3} - 82x^{2} - 4766x + 384036$
58.6.a.c	$x^{4} + 20x^{3} - 3096x^{2} - 120192x - 1082880$
58.6.a.d	$x^{4} - 106x^{3} - 2496x^{2} + 100560x + 100224$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(58))$	13
Old forms	22
Cusp forms $S_{6}(\Gamma_0(58))$	35
Eisenstein series $E_{6}(\Gamma_0(58))$	4
Modular forms $M_{6}(\Gamma_0(58))$	39

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(58)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.241 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, 29, 58]: [1, 2, 13, 39] (certified at ell = 2147483549)
T = [(3, 1), (5, 1)]; h of degree 13 = 1 x 13 (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(58, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 58:6 1000

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