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

Newform orbit 346.2.a.e

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 5q^{2} - 3q^{3} + 5q^{4} + 5q^{5} - 3q^{6} - 2q^{7} + 5q^{8} + 10q^{9} + 5q^{10} + 5q^{11} - 3q^{12} + 9q^{13} - 2q^{14} + 2q^{15} + 5q^{16} + 6q^{17} + 10q^{18} - 5q^{19} + 5q^{20} - 6q^{21} + 5q^{22} - 2q^{23} - 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{5} - 2x^{4} - 32x^{3} + 116x^{2} - 133x + 49$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
173	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	5	−3	5	−2	5	9	6	−5	−2	5	−8	7	12	−13	0	3	−1	17	−21	−4	−14	−2	−13	18	−6

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

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)[4]   # 346.2.a.e

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

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