Atlas › Modular forms › Weight 2 › Level 221 › 221.2.a › 221.2.a.f

Newform orbit 221.2.a.f

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 3q^{3} + 2q^{4} - 2q^{5} - 8q^{6} - 9q^{7} - 3q^{8} + 2q^{9} - 5q^{10} - 7q^{11} + q^{12} + 3q^{13} + 8q^{14} + 7q^{15} - 4q^{16} + 3q^{17} + 13q^{18} - 17q^{19} - 9q^{20} + q^{21} - 3q^{22} + 2q^{23} + 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{3} + 82x^{2} + 2108x + 16309$

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
$\operatorname{tr} a_p$	0	−3	−2	−9	−7	3	3	−17	2	4	−6	−4	2	6	−2	11	6	−15	−18	−20	4	−24	22	−17	1

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.185 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 13, 17, 221]: [0, 1, 2, 22] (certified at ell = 2147483137)
T = [(2, 1), (3, 3), (5, 5), (7, 7)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 4 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, 2)[5]   # 221.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 221:2 1000