Atlas › Modular forms › Weight 2 › Level 349 › 349.2.a › 349.2.a.b

Newform orbit 349.2.a.b

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$17q + 5q^{2} + 6q^{3} + 19q^{4} + 5q^{5} - 3q^{6} + q^{7} + 9q^{8} + 25q^{9} - 6q^{10} + 39q^{11} + 4q^{12} - 6q^{13} + 11q^{14} + 12q^{15} + 23q^{16} + q^{17} + 7q^{19} + 8q^{20} - 3q^{21} - q^{22} + 8q^{23} - 21q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 17. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{17} - 11x^{16} - x^{15} + 399x^{14} - 1051x^{13} - 4198x^{12} + 18687x^{11} + 11308x^{10} - 127618x^{9} + 50877x^{8} + 413606x^{7} - 340358x^{6} - 671456x^{5} + 673829x^{4} + 518852x^{3} - 511909x^{2} - 148020x + 94951$

Atkin–Lehner signs

$p$	$w_p$
349	$-1$

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	6	5	1	39	−6	1	7	8	9	−2	−7	3	1	16	27	74	−30	9	70	−38	7	47	23	−14

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.176 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, 349]: [0, 29] (certified at ell = 2147481901)
T = [(2, 1), (3, 1)]; h of degree 28 = 1 x 28 (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(349, 2)[1]   # 349.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 349:2 1000