Atlas › Modular forms › Weight 2 › Level 412 › 412.2.a

Newspace 412.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(412))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
412.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} - 2q^{5} - 3q^{7} + O(q^{8})$	−
412.2.a.b	2	$\mathbb{Q}(\sqrt{21})$	$2q - 2q^{3} - q^{5} + O(q^{8})$	−
412.2.a.c	4	degree 4	$4q + 2q^{3} + 3q^{5} + 5q^{7} + O(q^{8})$	+

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 $T$
412.2.a.a	$x^{2} + 8x - 4$
412.2.a.b	$x^{2} + 5x - 41$
412.2.a.c	$x^{4} - 11x^{3} - 65x^{2} + 822x - 1796$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(412))$	8
Old forms	42
Cusp forms $S_{2}(\Gamma_0(412))$	50
Eisenstein series $E_{2}(\Gamma_0(412))$	5
Modular forms $M_{2}(\Gamma_0(412))$	55

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(412)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(103))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(206))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.202 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, 103, 206, 412]: [0, 1, 2, 9, 28, 55] (certified at ell = 2147482501)
T = [(3, 1), (5, 3)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 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(412, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 412:2 1000

← 411.2.a · 413.2.a →