Atlas › Modular forms › Weight 2 › Level 422 › 422.2.a

Newspace 422.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(422))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
422.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} - 2q^{7} + O(q^{8})$	+ +	$-1$	422a1 rank 1
422.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 3q^{3} + 2q^{4} + 2q^{5} - 3q^{6} + 8q^{7} + O(q^{8})$	+ −	$+1$
422.2.a.c	3	degree 3	$3q - 3q^{2} - q^{3} + 3q^{4} - 5q^{5} + q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
422.2.a.d	3	degree 3	$3q - 3q^{2} - q^{3} + 3q^{4} + q^{5} + q^{6} + q^{7} + O(q^{8})$	+ −	$+1$
422.2.a.e	3	degree 3	$3q + 3q^{2} - 5q^{3} + 3q^{4} - 3q^{5} - 5q^{6} - 9q^{7} + O(q^{8})$	− −	$-1$
422.2.a.f	6	degree 6	$6q + 6q^{2} + 4q^{3} + 6q^{4} - 2q^{5} + 4q^{6} + 7q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ 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 $T$
422.2.a.a	$x + 28$
422.2.a.b	$x^{2} - 63x + 541$
422.2.a.c	$x^{3} - 22x^{2} - 579x + 125$
422.2.a.d	$x^{3} + 28x^{2} + 43x - 659$
422.2.a.e	$x^{3} + 94x^{2} + 1459x - 27103$
422.2.a.f	$x^{6} - 61x^{5} + 470x^{4} + 24232x^{3} - 263344x^{2} - 2609751x + 23349152$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(422))$	18
Old forms	34
Cusp forms $S_{2}(\Gamma_0(422))$	52
Eisenstein series $E_{2}(\Gamma_0(422))$	3
Modular forms $M_{2}(\Gamma_0(422))$	55

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(422)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(211))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.302 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, 211, 422]: [0, 1, 18, 55] (certified at ell = 2147482681)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 18 = 1 x 18 (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(422, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 422:2 1000

← 421.2.a · 423.2.a →