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

Newform orbit 111.4.a.f

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q + 3q^{2} + 21q^{3} + 45q^{4} + 14q^{5} + 9q^{6} + 43q^{7} + 57q^{8} + 63q^{9} - 24q^{10} - 31q^{11} + 135q^{12} + 135q^{13} - 139q^{14} + 42q^{15} + 261q^{16} - 31q^{17} + 27q^{18} + 249q^{19} - 102q^{20} + 129q^{21} + 43q^{22} + 155q^{23} + 171q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{7} - 17x^{6} - 542x^{5} + 10580x^{4} + 25047x^{3} - 936117x^{2} + 82026x + 23245110$

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
$\operatorname{tr} a_p$	3	21	14	43	−31	135	−31	249	155	134	54	259	−258	356	−974	−285	138	242	694	936	1937	830	−675	697	2022

$\operatorname{tr} a_n$ 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)[5]   # 111.4.a.f

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

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