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

Newform orbit 39.10.a.a

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 18q^{2} + 243q^{3} + 980q^{4} - 888q^{5} - 1458q^{6} - 14406q^{7} - 15480q^{8} + 19683q^{9} - 66044q^{10} - 91332q^{11} + 79380q^{12} + 85683q^{13} + 57456q^{14} - 71928q^{15} + 103184q^{16} + 582702q^{17} - 118098q^{18} - 618910q^{19} - 425544q^{20} - 1166886q^{21} - 2175400q^{22} + 1264632q^{23} - 1253880q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{3} + 906x^{2} - 973872x + 186071040$

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
$\operatorname{tr} a_p$	−18	243	−888	−14406	−91332	85683	582702	−618910	1264632	−9411426	−7847822	−26923130	−51617532	−27603940	19133916	8120034	23507568	36646558	−74652530	28518936	−445852854	103813272	243924216	−629566296	−1502912558

$\operatorname{tr} a_n$ 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)[0]   # 39.10.a.a

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

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