Atlas › Modular forms › Weight 4 › Level 221 › 221.4.a › 221.4.a.e

Newform orbit 221.4.a.e

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q + 5q^{2} + 20q^{3} + 49q^{4} + 18q^{5} - 38q^{6} + 94q^{7} + 6q^{8} + 100q^{9} + 65q^{10} + 164q^{11} + 137q^{12} + 182q^{13} - 40q^{14} + 70q^{15} + 293q^{16} + 238q^{17} + 208q^{18} + 364q^{19} + 203q^{20} + 298q^{21} + 7q^{22} + 30q^{23} - 103q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} - 25x^{13} + 31x^{12} + 3232x^{11} - 14654x^{10} - 154630x^{9} + 795858x^{8} + 3811992x^{7} - 16854655x^{6} - 56192157x^{5} + 150450235x^{4} + 478314968x^{3} - 363820404x^{2} - 1756237236x - 1159497628$

Atkin–Lehner signs

$p$	$w_p$
13	$-1$
17	$-1$

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
$\operatorname{tr} a_p$	5	20	18	94	164	182	238	364	30	412	786	426	238	626	492	−210	418	1618	1152	1412	922	−1892	−2966	250	1304

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.143 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, 13, 17, 221]: [1, 5, 6, 66] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 48 = 1 x 48 (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(221, 4)[4]   # 221.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 221:4 1000