Atlas › Modular forms › Weight 2 › Level 347 › 347.2.a › 347.2.a.d

Newform orbit 347.2.a.d

Trace form

The 19 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$19q + 7q^{3} + 22q^{4} + 10q^{5} - 6q^{6} + 9q^{7} - 3q^{8} + 22q^{9} + 12q^{10} + 10q^{12} + 39q^{13} - 5q^{14} + 2q^{15} + 20q^{16} + 11q^{17} - 13q^{18} + 5q^{19} + 3q^{20} + 6q^{22} - 21q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 19. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{19} - 7x^{18} - 39x^{17} + 358x^{16} + 349x^{15} - 6773x^{14} + 2782x^{13} + 62168x^{12} - 63594x^{11} - 300197x^{10} + 394450x^{9} + 780463x^{8} - 1077353x^{7} - 1146282x^{6} + 1343363x^{5} + 1032701x^{4} - 688395x^{3} - 496010x^{2} + 67126x + 56047$

Atkin–Lehner signs

$p$	$w_p$
347	$-1$

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
$\operatorname{tr} a_p$	0	7	10	9	0	39	11	5	0	7	14	50	−6	9	0	21	−21	39	11	−18	9	−17	−20	−3	72

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.26 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, 347]: [0, 30] (certified at ell = 2147481179)
T = [(2, 1), (3, 1)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 3 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(347, 2)[3]   # 347.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 347:2 1000