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

Newform orbit 221.2.a.d

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{21})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{2} + 43x + 415$

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
$\operatorname{tr} a_p$	−1	1	−2	−5	3	−2	2	5	6	18	8	−8	0	18	−2	−11	8	19	−18	4	8	−8	0	−9	7

$\operatorname{tr} a_n$ 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)[3]   # 221.2.a.d

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

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