Atlas › Modular forms › Weight 2 › Level 176 › 176.2.a

Newspace 176.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(176))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
176.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 3q^{5} - 2q^{7} + O(q^{8})$	−	176c1 rank 1
176.2.a.b	1	$\mathbb{Q}$	$q + q^{3} + q^{5} + 2q^{7} + O(q^{8})$	+	176b1 rank 0
176.2.a.c	1	$\mathbb{Q}$	$q + 3q^{3} - 3q^{5} + 2q^{7} + O(q^{8})$	−	176a1 rank 0
176.2.a.d	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{3} + 3q^{5} + 2q^{7} + O(q^{8})$	−

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 $T$
176.2.a.a	$x + 10$
176.2.a.b	$x - 4$
176.2.a.c	$x + 6$
176.2.a.d	$x^{2} - 8x - 52$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(176))$	5
Old forms	14
Cusp forms $S_{2}(\Gamma_0(176))$	19
Eisenstein series $E_{2}(\Gamma_0(176))$	11
Modular forms $M_{2}(\Gamma_0(176))$	30

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(176)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(44))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(88))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.209 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, 4, 8, 11, 16, 22, 44, 88, 176]: [0, 1, 2, 3, 2, 4, 5, 9, 16, 28] (certified at ell = 2147482921)
T = [(3, 1), (5, 3)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 1 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(176, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 176:2 1000

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