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

Newform orbit 39.8.a.c

Trace form

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

$4q - 6q^{2} - 108q^{3} + 406q^{4} - 276q^{5} + 162q^{6} - 1116q^{7} + 150q^{8} + 2916q^{9} - 3784q^{10} + 4968q^{11} - 10962q^{12} - 8788q^{13} + 65472q^{14} + 7452q^{15} + 87474q^{16} + 22944q^{17} - 4374q^{18} - 26628q^{19} + 161976q^{20} + 30132q^{21} + 76292q^{22} + 23712q^{23} - 4050q^{24} + O(q^{25})$

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

$x^{4} + 282x^{3} - 109713x^{2} - 20865612x + 954770284$

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
$\operatorname{tr} a_p$	−6	−108	−276	−1116	4968	−8788	22944	−26628	23712	227592	531380	−630032	212028	−1883816	−218976	4469064	6869472	458456	−2218700	2473128	10457616	5130864	16493520	−3209484	17620768

$\operatorname{tr} a_n$ 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)[2]   # 39.8.a.c

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

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