Atlas › Modular forms › Weight 2 › Level 353 › 353.2.a › 353.2.a.c

Newform orbit 353.2.a.c

Trace form

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

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

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

$x^{11} + 10x^{10} + 19x^{9} - 69x^{8} - 200x^{7} + 99x^{6} + 555x^{5} + 156x^{4} - 474x^{3} - 302x^{2} + 39x + 45$

Atkin–Lehner signs

$p$	$w_p$
353	$+1$

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	−5	−4	−25	−4	−13	−2	−18	−19	6	−18	−17	4	−45	−11	2	11	15	−36	4	−2	−33	−2	23	−13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.181 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, 353]: [0, 30] (certified at ell = 2147480897)
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(353, 2)[2]   # 353.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 353:2 1000