Atlas › Modular forms › Weight 2 › Level 346 › 346.2.a

Newspace 346.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(346))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
346.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 3q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$	346b1 rank 1
346.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$	346a1 rank 0
346.2.a.c	3	degree 3	$3q - 3q^{2} + q^{3} + 3q^{4} - q^{6} - 3q^{7} + O(q^{8})$	+ −	$+1$
346.2.a.d	4	degree 4	$4q - 4q^{2} - 2q^{3} + 4q^{4} - 5q^{5} + 2q^{6} - q^{7} + O(q^{8})$	+ +	$-1$
346.2.a.e	5	degree 5	$5q + 5q^{2} - 3q^{3} + 5q^{4} + 5q^{5} - 3q^{6} - 2q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
346.2.a.a	$x + 4$
346.2.a.b	$x$
346.2.a.c	$x^{3} - x^{2} - 15x + 28$
346.2.a.d	$x^{4} + 7x^{3} + 10x^{2} - 17x - 32$
346.2.a.e	$x^{5} - 2x^{4} - 32x^{3} + 116x^{2} - 133x + 49$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(346))$	14
Old forms	28
Cusp forms $S_{2}(\Gamma_0(346))$	42
Eisenstein series $E_{2}(\Gamma_0(346))$	3
Modular forms $M_{2}(\Gamma_0(346))$	45

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(346)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(173))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.188 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, 173, 346]: [0, 1, 15, 45] (certified at ell = 2147482093)
T = [(3, 1), (5, 1)]; h of degree 14 = 1 x 14 (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(346, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 346:2 1000

← 345.2.a · 347.2.a →