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

Newform orbit 939.2.a.e

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q - 3q^{2} - 10q^{3} + 9q^{4} - 8q^{5} + 3q^{6} + 6q^{7} - 12q^{8} + 10q^{9} - 5q^{11} - 9q^{12} - 11q^{13} - 13q^{14} + 8q^{15} - q^{16} - 14q^{17} - 3q^{18} - 14q^{19} - 13q^{20} - 6q^{21} + 12q^{22} + 12q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{10} + 11x^{9} + 17x^{8} - 144x^{7} - 340x^{6} + 573x^{5} + 1344x^{4} - 846x^{3} - 1621x^{2} + 426x + 575$

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
$\operatorname{tr} a_p$	−3	−10	−8	6	−5	−11	−14	−14	0	−29	−2	4	−23	20	−34	−13	−35	−53	20	−19	−9	28	−5	−18	9

$\operatorname{tr} a_n$ 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)[4]   # 939.2.a.e

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

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