Atlas › Modular forms › Weight 2 › Level 342 › 342.2.a

Newspace 342.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(342))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
342.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 2q^{5} + O(q^{8})$	+ +	342e1 rank 1
342.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 2q^{5} + O(q^{8})$	+ +	342f1 rank 0
342.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 4q^{7} + O(q^{8})$	+ −	342c1 rank 1
342.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 4q^{5} + 3q^{7} + O(q^{8})$	+ +	342g1 rank 0
342.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - q^{7} + O(q^{8})$	− −	342a1 rank 0
342.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 4q^{7} + O(q^{8})$	− −	342b1 rank 0
342.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{5} + O(q^{8})$	− +	342d1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{13} + 4T_{17} + 5T_{23} + 6T_{11} + 7T_{29} + 8T_{31}$ 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 $T$
342.2.a.a	$x + 68$
342.2.a.b	$x - 118$
342.2.a.c	$x + 40$
342.2.a.d	$x + 41$
342.2.a.e	$x + 73$
342.2.a.f	$x - 92$
342.2.a.g	$x - 12$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(342))$	7
Old forms	46
Cusp forms $S_{2}(\Gamma_0(342))$	53
Eisenstein series $E_{2}(\Gamma_0(342))$	15
Modular forms $M_{2}(\Gamma_0(342))$	68

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(342)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(38))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(57))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(114))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(171))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.423 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, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342]: [0, 1, 1, 3, 2, 5, 2, 7, 8, 24, 22, 64] (certified at ell = 2147483647)
T = [(5, 1), (7, 2), (13, 3), (17, 4), (23, 5), (11, 6), (29, 7), (31, 8)]; h of degree 7 = 1 x 7 (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(342, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 342:2 1000

← 341.2.a · 343.2.a →