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

Newform orbit 221.2.a.g

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{6} - 43x^{5} - 1569x^{4} + 85694x^{3} - 122721x^{2} - 29977475x + 289332409$

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	7	−1	6	−6	23	−10	−4	16	4	−4	10	−6	−27	10	11	18	0	12	−6	26	21	3

$\operatorname{tr} a_n$ 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)[6]   # 221.2.a.g

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

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