Atlas › Modular forms › Weight 4 › Level 186 › 186.4.a

Newspace 186.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(186))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
186.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 3q^{3} + 4q^{4} + 3q^{5} + 6q^{6} - 7q^{7} + O(q^{8})$	+ + −	$-1$
186.4.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + 3q^{3} + 4q^{4} - 11q^{5} - 6q^{6} - 22q^{7} + O(q^{8})$	+ − −	$+1$
186.4.a.c	1	$\mathbb{Q}$	$q - 2q^{2} + 3q^{3} + 4q^{4} - 11q^{5} - 6q^{6} + 9q^{7} + O(q^{8})$	+ − +	$-1$
186.4.a.d	1	$\mathbb{Q}$	$q - 2q^{2} + 3q^{3} + 4q^{4} + 15q^{5} - 6q^{6} + 17q^{7} + O(q^{8})$	+ − −	$+1$
186.4.a.e	1	$\mathbb{Q}$	$q + 2q^{2} - 3q^{3} + 4q^{4} - 7q^{5} - 6q^{6} - 3q^{7} + O(q^{8})$	− + +	$-1$
186.4.a.f	1	$\mathbb{Q}$	$q + 2q^{2} + 3q^{3} + 4q^{4} - 21q^{5} + 6q^{6} - 19q^{7} + O(q^{8})$	− − −	$-1$
186.4.a.g	1	$\mathbb{Q}$	$q + 2q^{2} + 3q^{3} + 4q^{4} - q^{5} + 6q^{6} - 6q^{7} + O(q^{8})$	− − +	$+1$
186.4.a.h	2	$\mathbb{Q}(\sqrt{97})$	$2q - 4q^{2} - 6q^{3} + 8q^{4} - 2q^{5} + 12q^{6} + 7q^{7} + O(q^{8})$	+ + +	$+1$
186.4.a.i	2	$\mathbb{Q}(\sqrt{521})$	$2q + 4q^{2} + 6q^{3} + 8q^{4} + 15q^{5} + 12q^{6} + 29q^{7} + O(q^{8})$	− − +	$+1$
186.4.a.j	3	degree 3	$3q + 6q^{2} - 9q^{3} + 12q^{4} + 8q^{5} - 18q^{6} + 11q^{7} + O(q^{8})$	− + −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
186.4.a.a	$x + 18$
186.4.a.b	$x + 77$
186.4.a.c	$x - 16$
186.4.a.d	$x - 66$
186.4.a.e	$x + 16$
186.4.a.f	$x + 78$
186.4.a.g	$x + 19$
186.4.a.h	$x^{2} - 19x - 8664$
186.4.a.i	$x^{2} - 102x + 2080$
186.4.a.j	$x^{3} - 41x^{2} + 96x + 6828$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(186))$	14
Old forms	78
Cusp forms $S_{4}(\Gamma_0(186))$	92
Eisenstein series $E_{4}(\Gamma_0(186))$	8
Modular forms $M_{4}(\Gamma_0(186))$	100

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(186)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(62))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(93))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.179 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, 3, 6, 31, 62, 93, 186]: [1, 2, 2, 5, 9, 26, 34, 100] (certified at ell = 2147482951)
T = [(5, 1), (7, 3)]; h of degree 14 = 1 x 14 (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(186, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 186:4 1000

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