Atlas › Modular forms › Weight 2 › Level 849 › 849.2.a › 849.2.a.e

Newform orbit 849.2.a.e

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q - q^{2} + 18q^{3} + 27q^{4} + q^{5} - q^{6} + 11q^{7} - 3q^{8} + 18q^{9} + 5q^{10} + 2q^{11} + 27q^{12} + 20q^{13} + q^{15} + 41q^{16} - 4q^{17} - q^{18} + 19q^{19} - 2q^{20} + 11q^{21} + 16q^{22} + 8q^{23} - 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{18} - 99x^{16} + 69x^{15} + 3831x^{14} - 5228x^{13} - 72326x^{12} + 147117x^{11} + 664353x^{10} - 1907242x^{9} - 2252578x^{8} + 11267716x^{7} - 3598036x^{6} - 24211609x^{5} + 28847420x^{4} - 418287x^{3} - 12547402x^{2} + 2322012x + 1774728$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
283	$+1$

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	18	1	11	2	20	−4	19	8	−5	42	36	3	14	29	−28	8	10	14	18	14	22	−9	10	16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.25 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, 283, 849]: [0, 1, 24, 96] (certified at ell = 2147480197)
T = [(2, 1), (5, 1)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 5 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(849, 2)[4]   # 849.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 849:2 1000