Atlas › Modular forms › Weight 2 › Level 232 › 232.2.a

Newspace 232.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(232))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
232.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 3q^{5} + 2q^{7} + O(q^{8})$	+	232a1 rank 1
232.2.a.b	1	$\mathbb{Q}$	$q + q^{3} + q^{5} + 2q^{7} + O(q^{8})$	+	232b1 rank 0
232.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} - 2q^{5} - 8q^{7} + O(q^{8})$	−
232.2.a.d	3	degree 3	$3q + 2q^{3} + 4q^{5} + O(q^{8})$	−

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 $T$
232.2.a.a	$x + 4$
232.2.a.b	$x - 2$
232.2.a.c	$x^{2} + 4x + 2$
232.2.a.d	$x^{3} - 6x^{2} + 2x + 4$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(232))$	7
Old forms	20
Cusp forms $S_{2}(\Gamma_0(232))$	27
Eisenstein series $E_{2}(\Gamma_0(232))$	7
Modular forms $M_{2}(\Gamma_0(232))$	34

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.276 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]: [0, 1, 2, 3, 3, 9, 18, 34] (certified at ell = 2147483549)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (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(232, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 232:2 1000

← 231.2.a · 233.2.a →