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

Newform orbit 221.4.a.d

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q - 4q^{2} + 4q^{3} + 68q^{4} - 2q^{5} + 98q^{6} + 36q^{7} + 21q^{8} + 175q^{9} + 63q^{10} - 56q^{11} + 63q^{12} - 169q^{13} - 134q^{15} + 484q^{16} - 221q^{17} + 355q^{18} + 376q^{19} - 93q^{20} + 150q^{21} - 561q^{22} + 16q^{23} + 869q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{13} - 447x^{11} - 503x^{10} + 65567x^{9} + 162129x^{8} - 3853597x^{7} - 13893445x^{6} + 79805668x^{5} + 327643179x^{4} - 629231976x^{3} - 2658459048x^{2} + 1682976960x + 5860946664$

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
$\operatorname{tr} a_p$	−4	4	−2	36	−56	−169	−221	376	16	108	588	−86	236	1386	−72	−184	522	1346	−372	−1830	−84	3982	1518	888	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 0.143 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)[3]   # 221.4.a.d

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

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