Atlas › Modular forms › Weight 8 › Level 58 › 58.8.a

Newspace 58.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(58))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
58.8.a.a	3	degree 3	$3q - 24q^{2} + 13q^{3} + 192q^{4} + 195q^{5} - 104q^{6} + 110q^{7} + O(q^{8})$	+ +	$+1$
58.8.a.b	3	degree 3	$3q + 24q^{2} - 25q^{3} + 192q^{4} - 305q^{5} - 200q^{6} - 1138q^{7} + O(q^{8})$	− +	$-1$
58.8.a.c	4	degree 4	$4q - 32q^{2} - 68q^{3} + 256q^{4} - 180q^{5} + 544q^{6} - 576q^{7} + O(q^{8})$	+ −	$-1$
58.8.a.d	5	degree 5	$5q + 40q^{2} + 56q^{3} + 320q^{4} + 570q^{5} + 448q^{6} + 920q^{7} + O(q^{8})$	− −	$+1$

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 $T$
58.8.a.a	$x^{3} - 208x^{2} - 135264x - 799848$
58.8.a.b	$x^{3} + 330x^{2} + 5508x + 23296$
58.8.a.c	$x^{4} + 248x^{3} - 142068x^{2} - 7826832x - 98075340$
58.8.a.d	$x^{5} - 626x^{4} - 74276x^{3} + 97828344x^{2} - 9268454508x - 1401159789576$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(58))$	15
Old forms	36
Cusp forms $S_{8}(\Gamma_0(58))$	51
Eisenstein series $E_{8}(\Gamma_0(58))$	4
Modular forms $M_{8}(\Gamma_0(58))$	55

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(58)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.299 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, 3, 19, 55] (certified at ell = 2147483549)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 5 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, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 58:8 1000

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