Atlas › Modular forms › Weight 2 › Level 341 › 341.2.a › 341.2.a.d

Newform orbit 341.2.a.d

Trace form

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

$11q + q^{2} + 4q^{3} + 19q^{4} + 3q^{5} - 5q^{6} + 5q^{7} - 3q^{8} + 23q^{9} + 3q^{10} + 11q^{11} - 5q^{12} - 2q^{13} + 6q^{14} + 3q^{15} + 35q^{16} - 10q^{17} - 26q^{18} + 24q^{19} + 10q^{20} - 11q^{21} + q^{22} + 2q^{23} - 36q^{24} + O(q^{25})$

Coefficient field

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

$x^{11} - 13x^{10} - 173x^{9} + 2819x^{8} + 4206x^{7} - 177570x^{6} + 358358x^{5} + 3031316x^{4} - 9365668x^{3} - 8491980x^{2} + 26721433x - 7025263$

Atkin–Lehner signs

$p$	$w_p$
11	$-1$
31	$+1$

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
$\operatorname{tr} a_p$	1	4	3	5	11	−2	−10	24	2	−12	−11	−12	−11	35	3	3	−4	−19	54	−7	−24	16	−3	15	2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.185 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, 11, 31, 341]: [0, 2, 3, 34] (certified at ell = 2147482951)
T = [(2, 1), (3, 3)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 4 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(341, 2)[3]   # 341.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 341:2 1000