Atlas › Modular forms › Weight 4 › Level 86 › 86.4.a

Newspace 86.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(86))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
86.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 8q^{3} + 4q^{4} + 6q^{5} - 16q^{6} + 14q^{7} + O(q^{8})$	+ +	$+1$
86.4.a.b	1	$\mathbb{Q}$	$q + 2q^{2} - 4q^{3} + 4q^{4} - 14q^{5} - 8q^{6} - 14q^{7} + O(q^{8})$	− +	$-1$
86.4.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q - 4q^{2} + 2q^{3} + 8q^{4} - 14q^{5} - 4q^{6} - 22q^{7} + O(q^{8})$	+ −	$-1$
86.4.a.d	3	degree 3	$3q - 6q^{2} - 9q^{3} + 12q^{4} - 5q^{5} + 18q^{6} - 8q^{7} + O(q^{8})$	+ +	$+1$
86.4.a.e	4	degree 4	$4q + 8q^{2} + 11q^{3} + 16q^{4} + q^{5} + 22q^{6} + 42q^{7} + O(q^{8})$	− −	$+1$

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 $T$
86.4.a.a	$x - 14$
86.4.a.b	$x + 18$
86.4.a.c	$x^{2} + 12x - 12$
86.4.a.d	$x^{3} + 14x^{2} - 461x - 6990$
86.4.a.e	$x^{4} - 12x^{3} - 153x^{2} + 1364x + 2364$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(86))$	11
Old forms	20
Cusp forms $S_{4}(\Gamma_0(86))$	31
Eisenstein series $E_{4}(\Gamma_0(86))$	4
Modular forms $M_{4}(\Gamma_0(86))$	35

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(86)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.106 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, 12, 35] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 2 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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 86:4 1000

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