Atlas › Modular forms › Weight 8 › Level 39 › 39.8.a › 39.8.a.b

Newform orbit 39.8.a.b

Trace form

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

$3q - 14q^{2} - 81q^{3} + 340q^{4} - 370q^{5} + 378q^{6} + 48q^{7} - 888q^{8} + 2187q^{9} + 4460q^{10} + 2920q^{11} - 9180q^{12} + 6591q^{13} - 49288q^{14} + 9990q^{15} - 18800q^{16} - 43354q^{17} - 10206q^{18} - 50756q^{19} - 130280q^{20} - 1296q^{21} - 73472q^{22} - 18408q^{23} + 23976q^{24} + O(q^{25})$

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

$x^{3} + 384x^{2} - 64344x - 11044800$

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
$\operatorname{tr} a_p$	−14	−81	−370	48	2920	6591	−43354	−50756	−18408	114554	93248	−363262	−556398	1393420	−938604	−1293270	−5846928	−6300502	3619508	−5742612	−6208602	−5789976	−1217504	−16081814	8275742

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.281 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, 3, 9, 34] (certified at ell = 2147483629)
T = [(2, 1), (5, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 5 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, 8)[1]   # 39.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 39:8 1000