Atlas › Modular forms › Weight 4 › Level 232 › 232.4.a

Newspace 232.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(232))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
232.4.a.a	3	degree 3	$3q + 3q^{3} + 11q^{5} - 38q^{7} + O(q^{8})$	+
232.4.a.b	3	degree 3	$3q + 6q^{3} + 4q^{5} + 16q^{7} + O(q^{8})$	+
232.4.a.c	4	degree 4	$4q - 20q^{5} - 8q^{7} + O(q^{8})$	−
232.4.a.d	5	degree 5	$5q + 4q^{3} + 10q^{5} + 32q^{7} + O(q^{8})$	−
232.4.a.e	6	degree 6	$6q - 5q^{3} - 5q^{5} - 38q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
232.4.a.a	$x^{3} - 14x^{2} - 292x + 3576$
232.4.a.b	$x^{3} - 10x^{2} - 332x + 3568$
232.4.a.c	$x^{4} + 20x^{3} - 20x^{2} - 1032x + 2004$
232.4.a.d	$x^{5} - 14x^{4} - 460x^{3} + 5600x^{2} - 732x - 2664$
232.4.a.e	$x^{6} + 10x^{5} - 232x^{4} - 2792x^{3} - 4016x^{2} + 23168x + 11008$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(232))$	21
Old forms	65
Cusp forms $S_{4}(\Gamma_0(232))$	86
Eisenstein series $E_{4}(\Gamma_0(232))$	8
Modular forms $M_{4}(\Gamma_0(232))$	94

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(232)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(58))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(116))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.153 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, 29, 58, 116, 232]: [1, 2, 3, 5, 9, 25, 48, 94] (certified at ell = 2147483549)
T = [(3, 1), (5, 1)]; h of degree 21 = 1 x 21 (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(232, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 232:4 1000

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