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

Newform orbit 346.2.a.d

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} + 7x^{3} + 10x^{2} - 17x - 32$

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 odd 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
$\operatorname{tr} a_p$	−4	−2	−5	−1	−13	3	−6	−12	−1	−21	−3	7	−16	−10	7	3	4	−3	−8	−29	18	−5	6	3	10

$\operatorname{tr} a_n$ 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)[3]   # 346.2.a.d

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

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