Atlas › Modular forms › Weight 2 › Level 428 › 428.2.a

Newspace 428.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(428))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
428.2.a.a	1	$\mathbb{Q}$	$q - q^{3} + 2q^{5} - 4q^{7} + O(q^{8})$	−	428b1 rank 1
428.2.a.b	1	$\mathbb{Q}$	$q + q^{3} + 2q^{5} + 4q^{7} + O(q^{8})$	+	428a1 rank 0
428.2.a.c	2	$\mathbb{Q}(\sqrt{13})$	$2q - 3q^{3} - q^{5} - 2q^{7} + O(q^{8})$	−
428.2.a.d	5	degree 5	$5q + 5q^{3} - q^{5} + 6q^{7} + O(q^{8})$	+

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 $T$
428.2.a.a	$x - 5$
428.2.a.b	$x - 7$
428.2.a.c	$x^{2} + 6x - 4$
428.2.a.d	$x^{5} - 2x^{4} - 203x^{3} + 257x^{2} + 8798x - 11236$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(428))$	9
Old forms	43
Cusp forms $S_{2}(\Gamma_0(428))$	52
Eisenstein series $E_{2}(\Gamma_0(428))$	5
Modular forms $M_{2}(\Gamma_0(428))$	57

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(428)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(107))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(214))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.203 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, 107, 214, 428]: [0, 1, 2, 10, 29, 57] (certified at ell = 2147482621)
T = [(3, 1), (5, 3)]; h of degree 9 = 1 x 9 (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(428, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 428:2 1000

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