Atlas › Modular forms › Weight 6 › Level 39 › 39.6.a › 39.6.a.c

Newform orbit 39.6.a.c

Trace form

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

$3q - 27q^{3} + 14q^{4} + 54q^{5} + 84q^{7} + 234q^{8} + 243q^{9} + 880q^{10} + 876q^{11} - 126q^{12} + 507q^{13} + 1896q^{14} - 486q^{15} - 1438q^{16} + 102q^{17} - 16q^{19} + 2124q^{20} - 756q^{21} - 2776q^{22} - 2106q^{24} + O(q^{25})$

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

$x^{3} - 54x^{2} - 3483x + 17496$

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
$\operatorname{tr} a_p$	0	−27	54	84	876	507	102	−16	0	9666	−10196	11818	35490	2780	25728	36786	−27516	45994	−42536	−54432	27846	−80568	24012	117450	−20930

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.232 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, 7, 26] (certified at ell = 2147483629)
T = [(2, 1), (5, 1)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 3 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, 6)[2]   # 39.6.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 39:6 1000