Atlas › Modular forms › Weight 6 › Level 59 › 59.6.a

Newspace 59.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(59))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
59.6.a.a	9	degree 9	$9q - 9q^{2} + 63q^{4} - 72q^{5} - 274q^{6} - 208q^{7} + O(q^{8})$	+	$-1$
59.6.a.b	15	degree 15	$15q + 3q^{2} + 18q^{3} + 303q^{4} + 128q^{5} + 14q^{6} + 282q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
59.6.a.a	$x^{9} + 9x^{8} - 1036x^{7} - 2081x^{6} + 270650x^{5} - 14210x^{4} - 20441539x^{3} + 10268630x^{2} + 327603856x - 270245344$
59.6.a.b	$x^{15} - 21x^{14} - 2818x^{13} + 45592x^{12} + 3256636x^{11} - 36054946x^{10} - 1958286259x^{9} + 12683401959x^{8} + 642288327357x^{7} - 1891527333549x^{6} - 113103468686273x^{5} + 37981626453747x^{4} + 9749892991189600x^{3} + 16196441633250386x^{2} - 306086191132525860x - 1023863300749578656$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(59))$	24
Old forms	0
Cusp forms $S_{6}(\Gamma_0(59))$	24
Eisenstein series $E_{6}(\Gamma_0(59))$	2
Modular forms $M_{6}(\Gamma_0(59))$	26

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.233 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, 59]: [1, 26] (certified at ell = 2147482481)
T = [(2, 1), (3, 1)]; h of degree 24 = 1 x 24 (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(59, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 59:6 1000

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