Atlas › Modular forms › Weight 2 › Level 343 › 343.2.a

Newspace 343.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(343))$

Newforms

Label	Dim	Coefficient field	Trace form	CM
343.2.a.a	3	degree 3	$3q - 4q^{2} + 4q^{4} + O(q^{8})$	$\mathbb{Q}(\sqrt{-7})$
343.2.a.b	3	degree 3	$3q + 3q^{2} + 11q^{4} + O(q^{8})$	$\mathbb{Q}(\sqrt{-7})$
343.2.a.c	6	degree 6	$6q - 2q^{2} - 5q^{3} + 4q^{4} - 11q^{5} - q^{6} + O(q^{8})$
343.2.a.d	6	degree 6	$6q - 2q^{2} + 5q^{3} + 4q^{4} + 11q^{5} + q^{6} + O(q^{8})$
343.2.a.e	6	degree 6	$6q + 4q^{2} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
343.2.a.a	$x^{3} + 4x^{2} + 3x - 1$
343.2.a.b	$x^{3} - 3x^{2} - 4x + 13$
343.2.a.c	$x^{6} + 7x^{5} + 4x^{4} - 28x^{3} - 4x^{2} + 14x - 1$
343.2.a.d	$x^{6} - 3x^{5} - 18x^{4} + 62x^{3} - 98x + 49$
343.2.a.e	$x^{6} - 4x^{5} - 18x^{4} + 50x^{3} + 105x^{2} - 98x - 49$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(343))$	24
Old forms	2
Cusp forms $S_{2}(\Gamma_0(343))$	26
Eisenstein series $E_{2}(\Gamma_0(343))$	13
Modular forms $M_{2}(\Gamma_0(343))$	39

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(343)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(49))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.182 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, 7, 49, 343]: [0, 1, 5, 33] (certified at ell = 2147479489)
T = [(2, 1), (3, 1)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 3 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(343, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 343:2 1000

← 342.2.a · 344.2.a →