Atlas › Modular forms › Weight 4 › Level 111 › 111.4.a › 111.4.a.e

Newform orbit 111.4.a.e

Trace form

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

$5q + 4q^{2} - 15q^{3} + 36q^{4} + 18q^{5} - 12q^{6} - 12q^{7} + 42q^{8} + 45q^{9} + 60q^{10} + 124q^{11} - 108q^{12} - 46q^{13} + 300q^{14} - 54q^{15} + 100q^{16} + 26q^{17} + 36q^{18} + 122q^{19} + 378q^{20} + 36q^{21} + 172q^{22} + 18q^{23} - 126q^{24} + O(q^{25})$

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

$x^{5} - 22x^{4} - 378x^{3} + 9652x^{2} - 46995x + 2764$

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 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	−15	18	−12	124	−46	26	122	18	−142	−330	−185	1014	−150	300	626	882	1154	196	−180	−1402	−1438	1568	2954	762

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.111 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, 2, 11, 40] (certified at ell = 2147483629)
T = [(2, 1), (5, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 3 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, 4)[4]   # 111.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 111:4 1000