Atlas › Modular forms › Weight 6 › Level 111 › 111.6.a › 111.6.a.a

Newform orbit 111.6.a.a

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q - 3q^{2} + 45q^{3} + 49q^{4} - 64q^{5} - 27q^{6} - 216q^{7} - 495q^{8} + 405q^{9} - 1020q^{10} - 598q^{11} + 441q^{12} - 1984q^{13} - 739q^{14} - 576q^{15} + 673q^{16} - 556q^{17} - 243q^{18} - 3628q^{19} + 1776q^{20} - 1944q^{21} - 1187q^{22} - 448q^{23} - 4455q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} + 67x^{4} - 3980x^{3} - 275284x^{2} + 3998052x + 279930168$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
37	$-1$

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	45	−64	−216	−598	−1984	−556	−3628	−448	−8704	−8416	6845	−29574	−44336	14212	−33138	41034	−88610	−3060	15708	−13844	21880	4782	89236	−164398

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.28 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 engine's checks

dim M^+ at levels [1, 3, 37, 111]: [1, 3, 17, 66] (certified at ell = 2147483629)
T = [(2, 1), (5, 1)]; h of degree 30 = 1 x 30 (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(111, 6)[0]   # 111.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 111:6 1000