Atlas › Modular forms › Weight 2 › Level 939 › 939.2.a › 939.2.a.d

Newform orbit 939.2.a.d

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - q^{2} + 6q^{3} + q^{4} - 6q^{5} - q^{6} - 6q^{7} + 6q^{9} - 12q^{10} - 3q^{11} + q^{12} - 23q^{13} - 5q^{14} - 6q^{15} - 5q^{16} - 14q^{17} - q^{18} - 14q^{19} - 3q^{20} - 6q^{21} + 2q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{6} + 7x^{5} + 9x^{4} - 34x^{3} - 103x^{2} - 90x - 25$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
313	$-1$

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
$\operatorname{tr} a_p$	−1	6	−6	−6	−3	−23	−14	−14	2	−7	−12	−4	7	−18	10	15	−3	−31	−24	−11	1	−24	−1	−8	21

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.375 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 3, 313, 939]: [0, 1, 26, 106] (certified at ell = 2147482273)
T = [(2, 1), (5, 1)]; h of degree 53 = 1 x 53 (phi(ord eps) x dim S_k^new); 6 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(939, 2)[3]   # 939.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 939:2 1000