Atlas › Modular forms › Weight 2 › Level 839 › 839.2.a › 839.2.a.b

Newform orbit 839.2.a.b

Trace form

The 51 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$51q + 4q^{2} + 4q^{3} + 66q^{4} + 6q^{5} + 15q^{6} + 19q^{7} + 9q^{8} + 81q^{9} + 8q^{10} + 18q^{11} + 4q^{12} + 22q^{13} + 2q^{14} + 9q^{15} + 88q^{16} + 19q^{17} + 8q^{18} + 32q^{19} - 10q^{20} + 29q^{21} + 6q^{22} + 4q^{23} + 31q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 51. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{51} - 8x^{50} - 184x^{49} + 1594x^{48} + 14939x^{47} - 143395x^{46} - 704484x^{45} + 7737806x^{44} + 21318920x^{43} - 280879638x^{42} - 428803527x^{41} + 7296002219x^{40} + 5625262188x^{39} - 140927977357x^{38} - 41350682615x^{37} + 2077149411818x^{36} - 6706742626x^{35} - 23782554759826x^{34} + 4118634889627x^{33} + 214103188531135x^{32} - 53545723412869x^{31} - 1526675277659980x^{30} + 392950917282387x^{29} + 8648472158274589x^{28} - 1845518590234454x^{27} - 38878596935085263x^{26} + 5201987689936533x^{25} + 137973009584234672x^{24} - 4503357483218454x^{23} - 382979104904462344x^{22} - 29732591701361711x^{21} + 820657955453733510x^{20} + 152943352497620590x^{19} - 1335234492910897604x^{18} - 380458788351567455x^{17} + 1617914988314212893x^{16} + 593437402877227888x^{15} - 1429144421232849303x^{14} - 610171340623036888x^{13} + 899058834879418524x^{12} + 413978858978943828x^{11} - 392382466288302294x^{10} - 180531619155741397x^{9} + 115125210830184059x^{8} + 47762245460970539x^{7} - 21687950851813216x^{6} - 6801940095658336x^{5} + 2374583616112294x^{4} + 375350374227811x^{3} - 111604956059785x^{2} + 2468858826330x - 13560307155$

Atkin–Lehner signs

$p$	$w_p$
839	$-1$

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
$\operatorname{tr} a_p$	4	4	6	19	18	22	19	32	4	26	31	42	49	24	−1	−5	7	74	13	6	110	75	−27	23	70

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.274 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, 839]: [0, 71] (certified at ell = 2147479751)
T = [(2, 1), (3, 1)]; h of degree 70 = 1 x 70 (phi(ord eps) x dim S_k^new); 7 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(839, 2)[1]   # 839.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 839:2 1000