Atlas › Modular forms › Weight 2 › Level 340 › 340.2.a

Newspace 340.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(340))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
340.2.a.a	1	$\mathbb{Q}$	$q - q^{5} - 4q^{7} + O(q^{8})$	+ −			340a1 rank 1
340.2.a.b	3	degree 3	$3q + 3q^{5} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7}$ 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 $T$
340.2.a.a	$x + 12$
340.2.a.b	$x^{3} - 128x + 256$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(340))$	4
Old forms	45
Cusp forms $S_{2}(\Gamma_0(340))$	49
Eisenstein series $E_{2}(\Gamma_0(340))$	11
Modular forms $M_{2}(\Gamma_0(340))$	60

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(340)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(68))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(85))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(170))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.207 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, 5, 10, 17, 20, 34, 68, 85, 170, 340]: [0, 1, 2, 1, 3, 2, 6, 6, 12, 10, 30, 60] (certified at ell = 2147483489)
T = [(3, 1), (7, 3)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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(340, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 340:2 1000

← 339.2.a · 341.2.a →