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

Newform orbit 939.2.a.g

Trace form

The 19 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$19q + q^{2} + 19q^{3} + 27q^{4} + 9q^{5} + q^{6} + 18q^{7} + 19q^{9} + 4q^{10} + q^{11} + 27q^{12} + 26q^{13} - 7q^{14} + 9q^{15} + 31q^{16} + 7q^{17} + q^{18} + 14q^{19} - 3q^{20} + 18q^{21} + 16q^{22} - 8q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 19. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{19} - 10x^{18} - 54x^{17} + 839x^{16} - 226x^{15} - 24231x^{14} + 56275x^{13} + 278244x^{12} - 1084982x^{11} - 1054183x^{10} + 8348837x^{9} - 1754104x^{8} - 31565290x^{7} + 22970355x^{6} + 60838989x^{5} - 59445832x^{4} - 54860511x^{3} + 56460085x^{2} + 16485649x - 11476187$

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 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
$\operatorname{tr} a_p$	1	19	9	18	1	26	7	14	−8	21	18	10	11	16	−8	−2	−16	63	2	−25	11	51	−13	5	34

$\operatorname{tr} a_n$ 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)[6]   # 939.2.a.g

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

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