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

Newform orbit 939.2.a.f

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q + 3q^{2} - 15q^{3} + 15q^{4} + 13q^{5} - 3q^{6} - 10q^{7} + 6q^{8} + 15q^{9} + 4q^{10} + 5q^{11} - 15q^{12} + 10q^{13} + 5q^{14} - 13q^{15} + 19q^{16} + 25q^{17} + 3q^{18} + 6q^{19} + 33q^{20} + 10q^{21} - 8q^{22} + 12q^{23} - 6q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{15} - 16x^{14} + 56x^{13} + 303x^{12} - 1953x^{11} - 1485x^{10} + 22241x^{9} - 1317x^{8} - 120033x^{7} + 10754x^{6} + 308680x^{5} + 43921x^{4} - 306979x^{3} - 82517x^{2} + 98019x + 30677$

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
$\operatorname{tr} a_p$	3	−15	13	−10	5	10	25	6	12	33	−2	4	23	−10	6	24	18	49	−24	15	11	−25	15	45	−10

$\operatorname{tr} a_n$ 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)[5]   # 939.2.a.f

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

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