Atlas › Modular forms › Weight 4 › Level 214 › 214.4.a › 214.4.a.d

Newform orbit 214.4.a.d

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 16q^{2} + 14q^{3} + 32q^{4} + 23q^{5} + 28q^{6} + 50q^{7} + 64q^{8} + 110q^{9} + 46q^{10} + 89q^{11} + 56q^{12} + 3q^{13} + 100q^{14} + 77q^{15} + 128q^{16} + 153q^{17} + 220q^{18} + 171q^{19} + 92q^{20} + 299q^{21} + 178q^{22} + 163q^{23} + 112q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{8} - 37x^{7} - 202x^{6} + 20159x^{5} - 160264x^{4} - 1367688x^{3} + 15815061x^{2} + 18487350x - 335982492$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
107	$-1$

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
$\operatorname{tr} a_p$	16	14	23	50	89	3	153	171	163	−111	298	−296	567	563	−295	−29	505	−872	−2866	603	284	−10	−282	1039	−1262

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.139 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, 2, 107, 214]: [1, 2, 28, 83] (certified at ell = 2147482621)
T = [(3, 1), (5, 1)]; h of degree 27 = 1 x 27 (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(214, 4)[3]   # 214.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 214:4 1000