Atlas › Modular forms › Weight 4 › Level 166 › 166.4.a

Newspace 166.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(166))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
166.4.a.a	1	$\mathbb{Q}$	$q + 2q^{2} - 5q^{3} + 4q^{4} + 8q^{5} - 10q^{6} - 31q^{7} + O(q^{8})$	− +	$-1$
166.4.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q + 4q^{2} - 2q^{3} + 8q^{4} - 24q^{5} - 4q^{6} - 6q^{7} + O(q^{8})$	− +	$-1$
166.4.a.c	5	degree 5	$5q - 10q^{2} + 2q^{3} + 20q^{4} - 22q^{5} - 4q^{6} + 14q^{7} + O(q^{8})$	+ −	$-1$
166.4.a.d	6	degree 6	$6q - 12q^{2} - q^{3} + 24q^{4} + 3q^{5} + 2q^{6} - 14q^{7} + O(q^{8})$	+ +	$+1$
166.4.a.e	7	degree 7	$7q + 14q^{2} + 8q^{3} + 28q^{4} + 9q^{5} + 16q^{6} + 33q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
166.4.a.a	$x - 19$
166.4.a.b	$x^{2} + 74x + 1337$
166.4.a.c	$x^{5} + 64x^{4} - 514x^{3} - 52581x^{2} - 18637x + 5910115$
166.4.a.d	$x^{6} - 8x^{5} - 4411x^{4} + 53695x^{3} + 3411428x^{2} - 32839283x - 221044262$
166.4.a.e	$x^{7} - 35x^{6} - 5613x^{5} + 213609x^{4} + 4672368x^{3} - 191170071x^{2} + 1404175262x - 207299713$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(166))$	21
Old forms	40
Cusp forms $S_{4}(\Gamma_0(166))$	61
Eisenstein series $E_{4}(\Gamma_0(166))$	4
Modular forms $M_{4}(\Gamma_0(166))$	65

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(166)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(83))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.13 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, 83, 166]: [1, 2, 22, 65] (certified at ell = 2147483323)
T = [(3, 1), (5, 3)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 5 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(166, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 166:4 1000

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