Atlas › Modular forms › Weight 4 › Level 176 › 176.4.a

Newspace 176.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(176))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
176.4.a.a	1	$\mathbb{Q}$	$q - 7q^{3} + 9q^{5} - 2q^{7} + O(q^{8})$	−
176.4.a.b	1	$\mathbb{Q}$	$q - 4q^{3} + 14q^{5} + 8q^{7} + O(q^{8})$	−
176.4.a.c	1	$\mathbb{Q}$	$q - q^{3} - 3q^{5} + 10q^{7} + O(q^{8})$	+
176.4.a.d	1	$\mathbb{Q}$	$q + q^{3} - 7q^{5} + 6q^{7} + O(q^{8})$	−
176.4.a.e	1	$\mathbb{Q}$	$q + 5q^{3} - 7q^{5} + 26q^{7} + O(q^{8})$	−
176.4.a.f	1	$\mathbb{Q}$	$q + 7q^{3} - 19q^{5} - 14q^{7} + O(q^{8})$	+
176.4.a.g	2	$\mathbb{Q}(\sqrt{97})$	$2q - 9q^{3} + 11q^{5} - 10q^{7} + O(q^{8})$	+
176.4.a.h	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} - 6q^{5} + 56q^{7} + O(q^{8})$	+
176.4.a.i	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} + 2q^{5} - 20q^{7} + O(q^{8})$	−
176.4.a.j	3	degree 3	$3q - 2q^{3} + 8q^{5} - 24q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{13} + 9T_{17}$ 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 $T$
176.4.a.a	$x + 332$
176.4.a.b	$x - 898$
176.4.a.c	$x - 306$
176.4.a.d	$x + 972$
176.4.a.e	$x - 892$
176.4.a.f	$x + 1038$
176.4.a.g	$x^{2} + 1620x + 642132$
176.4.a.h	$x^{2} + 116x - 749356$
176.4.a.i	$x^{2} + 648x + 104208$
176.4.a.j	$x^{3} - 2254x^{2} + 1483500x - 292631208$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(176))$	15
Old forms	51
Cusp forms $S_{4}(\Gamma_0(176))$	66
Eisenstein series $E_{4}(\Gamma_0(176))$	12
Modular forms $M_{4}(\Gamma_0(176))$	78

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(176)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 5}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(16))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(22))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(44))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(88))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.219 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]: [1, 2, 3, 5, 4, 8, 11, 21, 40, 76] (certified at ell = 2147482921)
T = [(3, 1), (5, 3), (7, 5), (13, 7), (17, 9)]; h of degree 15 = 1 x 15 (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(176, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 176:4 1000

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