Atlas › Modular forms › Weight 2 › Level 439 › 439.2.a › 439.2.a.b

Newform orbit 439.2.a.b

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} + 5x^{8} - 5x^{7} - 38x^{6} + x^{5} + 51x^{4} - 19x^{2} + 2x + 1$

Atkin–Lehner signs

$p$	$w_p$
439	$+1$

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
$\operatorname{tr} a_p$	−1	−4	−12	0	−18	2	−19	9	−13	−26	−10	−2	−28	−2	1	−1	3	−15	10	−21	17	−20	−6	−28	−15

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.19 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, 439]: [0, 37] (certified at ell = 2147481151)
T = [(2, 1), (3, 1)]; h of degree 36 = 1 x 36 (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(439, 2)[1]   # 439.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 439:2 1000