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

Newform orbit 353.2.a.d

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q + 4q^{2} - 2q^{3} + 16q^{4} - 4q^{5} - 8q^{6} + 29q^{7} + 3q^{8} + 14q^{9} + 15q^{10} - 7q^{11} - 4q^{12} + 2q^{13} + 8q^{14} + 16q^{15} + 12q^{16} + 8q^{17} + 5q^{18} + 14q^{19} - 5q^{20} + 6q^{21} + 25q^{22} + 16q^{23} - 23q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} - 2x^{13} - 40x^{12} + 74x^{11} + 532x^{10} - 926x^{9} - 3025x^{8} + 4996x^{7} + 7374x^{6} - 11457x^{5} - 6816x^{4} + 9682x^{3} + 2239x^{2} - 2219x - 263$

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 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
$\operatorname{tr} a_p$	4	−2	−4	29	−7	2	8	14	16	−9	28	8	14	11	14	−27	−25	−3	37	3	14	39	9	0	24

$\operatorname{tr} a_n$ 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)[3]   # 353.2.a.d

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

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