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

Newform orbit 221.4.a.c

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q + q^{2} - 16q^{3} + 37q^{4} - 6q^{5} - 40q^{6} - 18q^{7} - 42q^{8} + 24q^{9} - 47q^{10} - 96q^{11} - 151q^{12} - 130q^{13} - 208q^{14} - 50q^{15} + 109q^{16} + 170q^{17} + 300q^{18} - 400q^{19} - 21q^{20} - 118q^{21} - 525q^{22} - 446q^{23} - 407q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} + 15x^{9} - 53x^{8} - 1456x^{7} - 1274x^{6} + 37230x^{5} + 37994x^{4} - 345288x^{3} - 5847x^{2} + 668155x - 107365$

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 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
$\operatorname{tr} a_p$	1	−16	−6	−18	−96	−130	170	−400	−446	−168	−282	−122	−526	−1334	120	−198	426	−1586	−216	1536	−970	−1236	−1302	1386	4492

$\operatorname{tr} a_n$ 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)[2]   # 221.4.a.c

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

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