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

Newform orbit 221.2.a.e

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{2} - 24x + 99$

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
$\operatorname{tr} a_p$	0	2	−2	4	4	−2	2	4	−6	−12	0	10	10	−12	−4	0	−4	8	0	4	−14	14	8	−4	−18

$\operatorname{tr} a_n$ 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)[4]   # 221.2.a.e

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

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