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

Newform orbit 389.2.a.d

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - 3q^{2} - 5q^{3} + q^{4} + 3q^{5} - q^{6} - 4q^{7} - 9q^{8} - q^{9} - 5q^{10} - 2q^{11} + 2q^{12} - 5q^{13} + 8q^{14} - 14q^{15} + 3q^{16} - 16q^{17} + 14q^{18} - 33q^{19} + 12q^{20} - 5q^{21} - 7q^{22} + 5q^{23} + 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{6} + 18x^{5} + 82x^{4} - 248x^{3} - 3034x^{2} - 8381x - 7489$

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 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
$\operatorname{tr} a_p$	−3	−5	3	−4	−2	−5	−16	−33	5	−14	−14	−3	−2	−15	2	26	1	−34	13	14	6	−13	−31	−6	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 0.187 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)[3]   # 389.2.a.d

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

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