Atlas › Modular forms › Weight 10 › Level 39 › 39.10.a › 39.10.a.d

Newform orbit 39.10.a.d

Trace form

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

$6q - 2q^{2} + 486q^{3} + 1854q^{4} + 444q^{5} - 162q^{6} + 5868q^{7} + 28638q^{8} + 39366q^{9} + 135720q^{10} - 13204q^{11} + 150174q^{12} - 171366q^{13} - 135016q^{14} + 35964q^{15} + 116802q^{16} + 70180q^{17} - 13122q^{18} + 820500q^{19} - 65936q^{20} + 475308q^{21} + 3253188q^{22} + 3039544q^{23} + 2319678q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{6} - 442x^{5} - 10154077x^{4} + 6108636384x^{3} + 25505657329180x^{2} - 20638379351215696x - 1121855873084983360$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
13	$+1$

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
$\operatorname{tr} a_p$	−2	486	444	5868	−13204	−171366	70180	820500	3039544	5533372	10389252	6228012	45176092	50615760	−35820324	−94984684	−267443348	98858652	−48267372	−921228764	25791132	1367790768	1044952004	1210026684	1008660156

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.336 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 engine's checks

dim M^+ at levels [1, 3, 13, 39]: [1, 4, 11, 44] (certified at ell = 2147483629)
T = [(2, 1), (5, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 7 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(39, 10)[3]   # 39.10.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 39:10 1000