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

Newform orbit 439.2.a.c

Trace form

The 25 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$25q + 4q^{2} + 3q^{3} + 28q^{4} + 15q^{5} + 6q^{8} + 52q^{9} - 8q^{10} + 24q^{11} - 6q^{12} + q^{13} + 17q^{14} + 5q^{15} + 30q^{16} + 25q^{17} - 3q^{18} - q^{19} + 25q^{20} - 3q^{21} - 10q^{22} + 26q^{23} - 9q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 25. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{25} - 7x^{24} - 78x^{23} + 638x^{22} + 2078x^{21} - 22652x^{20} - 19720x^{19} + 416374x^{18} - 68307x^{17} - 4423461x^{16} + 2881060x^{15} + 28571527x^{14} - 23685149x^{13} - 114991163x^{12} + 89171181x^{11} + 291131720x^{10} - 156851208x^{9} - 451620211x^{8} + 93443707x^{7} + 384331295x^{6} + 38723209x^{5} - 140850851x^{4} - 46388365x^{3} + 9213266x^{2} + 5123490x + 431903$

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 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
$\operatorname{tr} a_p$	4	3	15	0	24	1	25	−1	26	49	−11	4	42	−11	−12	36	−2	2	−23	31	2	−15	−22	84	−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 0.19 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)[2]   # 439.2.a.c

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

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