Atlas › Modular forms › Weight 4 › Level 218 › 218.4.a › 218.4.a.b

Newform orbit 218.4.a.b

Trace form

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

$6q - 12q^{2} - 8q^{3} + 24q^{4} + 13q^{5} + 16q^{6} - 45q^{7} - 48q^{8} + 36q^{9} - 26q^{10} - 18q^{11} - 32q^{12} - 51q^{13} + 90q^{14} + 24q^{15} + 96q^{16} - 43q^{17} - 72q^{18} - 70q^{19} + 52q^{20} - 17q^{21} + 36q^{22} - 26q^{23} + 64q^{24} + O(q^{25})$

Coefficient field

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

$x^{6} - 5x^{5} - 529x^{4} + 1173x^{3} + 6476x^{2} - 18572x + 8648$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
109	$-1$

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
$\operatorname{tr} a_p$	−12	−8	13	−45	−18	−51	−43	−70	−26	−123	−429	−850	142	−1403	−609	−331	821	−383	−381	834	−927	−28	−1513	1666	−2249

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.137 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, 109, 218]: [1, 2, 29, 85] (certified at ell = 2147482801)
T = [(3, 1), (5, 1)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 5 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(218, 4)[1]   # 218.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 218:4 1000