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

Newform orbit 39.10.a.b

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 36q^{2} - 324q^{3} + 472q^{4} - 1044q^{5} - 2916q^{6} + 5868q^{7} + 37872q^{8} + 26244q^{9} - 98608q^{10} - 44712q^{11} - 38232q^{12} - 114244q^{13} - 20448q^{14} + 84564q^{15} + 312096q^{16} - 1058688q^{17} + 236196q^{18} - 628476q^{19} - 3378528q^{20} - 475308q^{21} - 4870984q^{22} - 760032q^{23} - 3067632q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{4} + 1008x^{3} - 4726908x^{2} - 120239568x + 2799291354304$

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 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
$\operatorname{tr} a_p$	36	−324	−1044	5868	−44712	−114244	−1058688	−628476	−760032	−2079000	−7664644	6312784	−16794900	23208424	−59149728	81609192	38113056	−28763272	137926540	−426885768	−144117456	−380731824	−654601680	−1484607996	−2344214144

$\operatorname{tr} a_n$ 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)[1]   # 39.10.a.b

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

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