Atlas › Modular forms › Weight 2 › Level 344 › 344.2.a

Newspace 344.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(344))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
344.2.a.a	1	$\mathbb{Q}$	$q - 2q^{5} - 2q^{7} + O(q^{8})$	−	344a1 rank 1
344.2.a.b	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} - 2q^{5} - 2q^{7} + O(q^{8})$	+
344.2.a.c	3	degree 3	$3q + 3q^{3} + q^{5} + 6q^{7} + O(q^{8})$	−
344.2.a.d	5	degree 5	$5q - q^{3} + q^{5} + 2q^{7} + O(q^{8})$	+

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 $T$
344.2.a.a	$x + 6$
344.2.a.b	$x^{2} + 8x + 4$
344.2.a.c	$x^{3} - 6x^{2} - 49x + 82$
344.2.a.d	$x^{5} - 2x^{4} - 205x^{3} + 814x^{2} + 8556x - 40648$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(344))$	11
Old forms	30
Cusp forms $S_{2}(\Gamma_0(344))$	41
Eisenstein series $E_{2}(\Gamma_0(344))$	7
Modular forms $M_{2}(\Gamma_0(344))$	48

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(344)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(86))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(172))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.197 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, 8, 43, 86, 172, 344]: [0, 1, 2, 3, 4, 13, 25, 48] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 11 = 1 x 11 (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(344, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 344:2 1000

← 343.2.a · 345.2.a →