Atlas › Modular forms › Weight 2 › Level 239 › 239.2.a › 239.2.a.b

Newform orbit 239.2.a.b

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$17q + 3q^{3} + 22q^{4} + 6q^{5} + 2q^{6} + 5q^{7} - 3q^{8} + 28q^{9} + 5q^{10} - q^{11} - 6q^{12} + 15q^{13} - 3q^{14} - q^{15} + 24q^{16} + 4q^{17} - 13q^{18} + 24q^{19} + 4q^{20} + 9q^{21} - 10q^{22} - 9q^{23} - 5q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 17. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{17} - 3x^{16} - 65x^{15} + 214x^{14} + 1557x^{13} - 5760x^{12} - 16571x^{11} + 73712x^{10} + 70240x^{9} - 469977x^{8} - 6891x^{7} + 1468432x^{6} - 659353x^{5} - 2067403x^{4} + 1258919x^{3} + 1183921x^{2} - 575531x - 241031$

Atkin–Lehner signs

$p$	$w_p$
239	$-1$

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
$\operatorname{tr} a_p$	0	3	6	5	−1	15	4	24	−9	−2	28	11	20	−9	−18	−12	1	24	16	12	30	−10	−16	65	87

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.169 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, 239]: [0, 21] (certified at ell = 2147482093)
T = [(2, 1), (3, 1)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 2 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(239, 2)[1]   # 239.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 239:2 1000