Atlas › Modular forms › Weight 6 › Level 86 › 86.6.a

Newspace 86.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(86))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
86.6.a.a	3	degree 3	$3q - 12q^{2} + 8q^{3} + 48q^{4} - 14q^{5} - 32q^{6} - 74q^{7} + O(q^{8})$	+ +	$-1$
86.6.a.b	3	degree 3	$3q + 12q^{2} - 28q^{3} + 48q^{4} - 14q^{5} - 112q^{6} - 182q^{7} + O(q^{8})$	− −	$-1$
86.6.a.c	5	degree 5	$5q - 20q^{2} - q^{3} + 80q^{4} + 61q^{5} + 4q^{6} + 122q^{7} + O(q^{8})$	+ −	$+1$
86.6.a.d	6	degree 6	$6q + 24q^{2} + 17q^{3} + 96q^{4} + 61q^{5} + 68q^{6} + 210q^{7} + O(q^{8})$	− +	$+1$

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 $T$
86.6.a.a	$x^{3} + 6x^{2} - 932x - 7128$
86.6.a.b	$x^{3} + 42x^{2} - 1068x + 5256$
86.6.a.c	$x^{5} - 60x^{4} - 10011x^{3} + 713162x^{2} + 8135412x - 893359800$
86.6.a.d	$x^{6} - 78x^{5} - 12971x^{4} + 1153300x^{3} - 6315744x^{2} - 121153904x - 267365904$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(86))$	17
Old forms	36
Cusp forms $S_{6}(\Gamma_0(86))$	53
Eisenstein series $E_{6}(\Gamma_0(86))$	4
Modular forms $M_{6}(\Gamma_0(86))$	57

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(86)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.26 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, 43, 86]: [1, 2, 20, 57] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 17 = 1 x 17 (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(86, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 86:6 1000

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