Atlas › Modular forms › Weight 4 › Level 171 › 171.4.a

Newspace 171.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(171))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
171.4.a.a	1	$\mathbb{Q}$	$q - 3q^{2} + q^{4} + 6q^{5} - 16q^{7} + O(q^{8})$	−
171.4.a.b	1	$\mathbb{Q}$	$q + q^{2} - 7q^{4} + 12q^{5} - 20q^{7} + O(q^{8})$	+
171.4.a.c	1	$\mathbb{Q}$	$q + 3q^{2} + q^{4} - 6q^{5} - 16q^{7} + O(q^{8})$	−
171.4.a.d	1	$\mathbb{Q}$	$q + 3q^{2} + q^{4} + 12q^{5} + 11q^{7} + O(q^{8})$	−
171.4.a.e	2	$\mathbb{Q}(\sqrt{33})$	$2q - q^{2} + q^{4} - 22q^{5} + 36q^{7} + O(q^{8})$	+
171.4.a.f	3	degree 3	$3q - 3q^{2} + 21q^{4} - 14q^{5} - 35q^{7} + O(q^{8})$	+
171.4.a.g	3	degree 3	$3q + 3q^{2} + 9q^{4} + 12q^{5} - 18q^{7} + O(q^{8})$	−
171.4.a.h	4	degree 4	$4q - 3q^{2} + 25q^{4} + 6q^{5} + 38q^{7} + O(q^{8})$	−
171.4.a.i	6	degree 6	$6q + 42q^{4} + 20q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
171.4.a.a	$x - 3$
171.4.a.b	$x - 13$
171.4.a.c	$x + 3$
171.4.a.d	$x - 15$
171.4.a.e	$x^{2} + 23x + 58$
171.4.a.f	$x^{3} + 17x^{2} + 41x + 13$
171.4.a.g	$x^{3} - 15x^{2} - 75x + 285$
171.4.a.h	$x^{4} - 3x^{3} - 333x^{2} + 489x + 26886$
171.4.a.i	$x^{6} - 927x^{4} + 217275x^{2} - 1156381$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(171))$	22
Old forms	34
Cusp forms $S_{4}(\Gamma_0(171))$	56
Eisenstein series $E_{4}(\Gamma_0(171))$	8
Modular forms $M_{4}(\Gamma_0(171))$	64

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(171)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(57))^{\oplus 2}$

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, 3, 9, 19, 57, 171]: [1, 2, 4, 6, 22, 62] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 4 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(171, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 171:4 1000

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