Atlas › Modular forms › Weight 2 › Level 166 › 166.2.a

Newspace 166.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(166))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
166.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - 2q^{5} + q^{6} + q^{7} + O(q^{8})$	+ +	$-1$	166a1 rank 1
166.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 3q^{5} + 2q^{6} - 3q^{7} + O(q^{8})$	+ −	$+1$
166.2.a.c	3	degree 3	$3q + 3q^{2} + q^{3} + 3q^{4} - q^{5} + q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ 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 $T$
166.2.a.a	$x + 37$
166.2.a.b	$x^{2} - 34x + 269$
166.2.a.c	$x^{3} - 43x^{2} - 781x + 5759$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(166))$	6
Old forms	14
Cusp forms $S_{2}(\Gamma_0(166))$	20
Eisenstein series $E_{2}(\Gamma_0(166))$	3
Modular forms $M_{2}(\Gamma_0(166))$	23

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.177 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]: [0, 1, 8, 23] (certified at ell = 2147483323)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 6 = 1 x 6 (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(166, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 166:2 1000

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