Atlas › Modular forms › Weight 2 › Level 332 › 332.2.a

Newspace 332.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(332))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
332.2.a.a	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} - 6q^{7} + O(q^{8})$	−
332.2.a.b	2	$\mathbb{Q}(\sqrt{7})$	$2q - 2q^{5} + O(q^{8})$	+
332.2.a.c	3	degree 3	$3q + 4q^{3} + 2q^{5} + 8q^{7} + O(q^{8})$	+

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 $T$
332.2.a.a	$x^{2} + 2x - 7$
332.2.a.b	$x^{2} + 6x - 103$
332.2.a.c	$x^{3} - 10x^{2} - 39x + 139$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(332))$	7
Old forms	33
Cusp forms $S_{2}(\Gamma_0(332))$	40
Eisenstein series $E_{2}(\Gamma_0(332))$	5
Modular forms $M_{2}(\Gamma_0(332))$	45

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(332)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(83))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(166))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.186 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, 83, 166, 332]: [0, 1, 2, 8, 23, 45] (certified at ell = 2147483323)
T = [(3, 1), (5, 3)]; 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(332, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 332:2 1000

← 331.2.a · 333.2.a →