Atlas › Modular forms › Weight 8 › Level 22 › 22.8.a › 22.8.a.d

Newform orbit 22.8.a.d

Trace form

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

$2q + 16q^{2} - 23q^{3} + 128q^{4} + 331q^{5} - 184q^{6} + 1794q^{7} + 1024q^{8} + 3331q^{9} + 2648q^{10} + 2662q^{11} - 1472q^{12} - 5406q^{13} + 14352q^{14} - 11247q^{15} + 8192q^{16} + 15032q^{17} + 26648q^{18} + 16916q^{19} + 21184q^{20} - 124798q^{21} + 21296q^{22} - 51351q^{23} - 11776q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{14881})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{2} - 970x + 220344$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
11	$-1$

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
$\operatorname{tr} a_p$	16	−23	331	1794	2662	−5406	15032	16916	−51351	−207130	−19071	351333	123610	−159822	451160	−1260832	887547	−597918	2864711	1306267	−4577530	−2946342	9965450	10185377	−27765477

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.265 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, 2, 11, 22]: [1, 3, 8, 23] (certified at ell = 2147483171)
T = [(3, 1), (5, 3)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 2 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(22, 8)[3]   # 22.8.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 22:8 1000