Atlas › Modular forms › Weight 2 › Level 187 › 187.2.a

Newspace 187.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(187))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
187.2.a.a	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + 3q^{5} + 2q^{7} + O(q^{8})$	− +	$+1$	187a1 rank 0
187.2.a.b	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} + 4q^{5} - 5q^{7} + O(q^{8})$	+ −	$+1$	187b1 rank 0
187.2.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} + 4q^{4} - 4q^{5} - 6q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
187.2.a.d	2	$\mathbb{Q}(\sqrt{17})$	$2q + 4q^{2} - q^{3} + 4q^{4} + q^{5} - 2q^{6} + 3q^{7} + O(q^{8})$	− +	$+1$
187.2.a.e	3	degree 3	$3q - 2q^{2} - 3q^{3} + 2q^{4} - 7q^{5} + 4q^{6} + O(q^{8})$	+ +	$-1$
187.2.a.f	4	degree 4	$4q + q^{2} + q^{3} + 5q^{4} + 3q^{5} - 4q^{6} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
187.2.a.a	$x - 3$
187.2.a.b	$x - 2$
187.2.a.c	$x^{2} + 2x - 11$
187.2.a.d	$x^{2} - x - 38$
187.2.a.e	$x^{3} + 11x^{2} - 5x - 107$
187.2.a.f	$x^{4} - 4x^{3} - 90x^{2} + 344x + 173$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(187))$	13
Old forms	4
Cusp forms $S_{2}(\Gamma_0(187))$	17
Eisenstein series $E_{2}(\Gamma_0(187))$	3
Modular forms $M_{2}(\Gamma_0(187))$	20

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(187)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.174 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, 11, 17, 187]: [0, 2, 2, 20] (certified at ell = 2147482801)
T = [(2, 1), (3, 3)]; h of degree 13 = 1 x 13 (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(187, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 187:2 1000

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