Atlas › Modular forms › Weight 2 › Level 389 › 389.2.a › 389.2.a.e

Newform orbit 389.2.a.e

Trace form

The 20 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$20q + 3q^{2} + 11q^{3} + 27q^{4} + q^{5} + q^{6} + 12q^{7} + 3q^{8} + 23q^{9} + 7q^{10} + 10q^{11} + 16q^{12} + 17q^{13} - 10q^{14} - 10q^{15} + 41q^{16} + 2q^{17} + 2q^{18} + 51q^{19} - 14q^{20} - 11q^{21} - 19q^{22} + 3q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 20. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{20} - 36x^{19} + 238x^{18} + 5938x^{17} - 89638x^{16} - 98459x^{15} + 8381042x^{14} - 31945454x^{13} - 300466488x^{12} + 2266802048x^{11} + 2323820867x^{10} - 58089218087x^{9} + 92671844985x^{8} + 561570921194x^{7} - 1869681316392x^{6} - 683117287589x^{5} + 8599815908447x^{4} - 6905582251608x^{3} - 6096196967352x^{2} + 5699822108448x + 21372607552$

Atkin–Lehner signs

$p$	$w_p$
389	$-1$

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
$\operatorname{tr} a_p$	3	11	1	12	10	17	2	51	3	0	36	9	16	29	2	−20	29	46	21	−14	0	47	9	−10	11

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.187 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, 389]: [0, 33] (certified at ell = 2147479897)
T = [(2, 1), (3, 3)]; h of degree 32 = 1 x 32 (phi(ord eps) x dim S_k^new); 5 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(389, 2)[4]   # 389.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 389:2 1000