Atlas › Modular forms › Weight 4 › Level 187 › 187.4.a

Newspace 187.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(187))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
187.4.a.a	1	$\mathbb{Q}$	$q + q^{2} + 4q^{3} - 7q^{4} + 6q^{5} + 4q^{6} - 24q^{7} + O(q^{8})$	+ −	$-1$
187.4.a.b	3	degree 3	$3q - 9q^{2} + q^{3} + 3q^{4} - 19q^{5} - 3q^{6} + 9q^{7} + O(q^{8})$	+ −	$-1$
187.4.a.c	4	degree 4	$4q + 2q^{2} - 6q^{3} + 24q^{4} - 10q^{5} - 18q^{6} + 16q^{7} + O(q^{8})$	+ −	$-1$
187.4.a.d	10	degree 10	$10q - 8q^{2} - 9q^{3} + 40q^{4} - 41q^{5} + 31q^{6} - 63q^{7} + O(q^{8})$	− +	$-1$
187.4.a.e	10	degree 10	$10q + 4q^{2} + 11q^{3} + 40q^{4} + 47q^{5} + 105q^{6} - 17q^{7} + O(q^{8})$	+ +	$+1$
187.4.a.f	12	degree 12	$12q + 10q^{2} + 3q^{3} + 60q^{4} + 29q^{5} - 43q^{6} + 39q^{7} + O(q^{8})$	− −	$+1$

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 $T$
187.4.a.a	$x - 5$
187.4.a.b	$x^{3} + 8x^{2} - 49x - 95$
187.4.a.c	$x^{4} + 4x^{3} - 24x^{2} - 116x - 109$
187.4.a.d	$x^{10} + 17x^{9} - 112x^{8} - 2557x^{7} - 1011x^{6} + 70527x^{5} + 14443x^{4} - 548647x^{3} - 12582x^{2} + 1194836x - 59171$
187.4.a.e	$x^{10} - 15x^{9} - 241x^{8} + 3856x^{7} + 17756x^{6} - 301625x^{5} - 714331x^{4} + 9581074x^{3} + 18879033x^{2} - 105713102x - 220475646$
187.4.a.f	$x^{12} - 13x^{11} - 172x^{10} + 2291x^{9} + 10131x^{8} - 128357x^{7} - 280591x^{6} + 2604463x^{5} + 4391790x^{4} - 19719558x^{3} - 29372545x^{2} + 49234486x + 67321498$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(187))$	40
Old forms	12
Cusp forms $S_{4}(\Gamma_0(187))$	52
Eisenstein series $E_{4}(\Gamma_0(187))$	4
Modular forms $M_{4}(\Gamma_0(187))$	56

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(187)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.131 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]: [1, 4, 6, 56] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 40 = 1 x 40 (phi(ord eps) x dim S_k^new); 6 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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 187:4 1000

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