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

Newform orbit 349.2.a.a

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q - 5q^{2} - 6q^{3} + 5q^{4} - 9q^{5} - 5q^{6} - 3q^{7} - 15q^{8} + 3q^{9} + 2q^{10} - 31q^{11} - 4q^{13} - 7q^{14} - 12q^{15} + 5q^{16} - q^{17} - 17q^{19} - 10q^{20} - 15q^{21} + 17q^{22} - 24q^{23} - 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{11} + 11x^{10} + 34x^{9} - 14x^{8} - 218x^{7} - 265x^{6} + 52x^{5} + 240x^{4} + 118x^{3} + 8x^{2} - 6x - 1$

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 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
$\operatorname{tr} a_p$	−5	−6	−9	−3	−31	−4	−1	−17	−24	−17	−10	−1	−15	−5	4	−3	−52	0	−23	−30	12	11	−13	−19	26

$\operatorname{tr} a_n$ 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)[0]   # 349.2.a.a

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

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