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

Newform orbit 218.4.a.a

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 10q^{2} - 10q^{3} + 20q^{4} - 28q^{5} - 20q^{6} - 37q^{7} + 40q^{8} - 9q^{9} - 56q^{10} - 65q^{11} - 40q^{12} - 161q^{13} - 74q^{14} - 74q^{15} + 80q^{16} - 145q^{17} - 18q^{18} - 151q^{19} - 112q^{20} - 197q^{21} - 130q^{22} - 251q^{23} - 80q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} + 38x^{4} + 493x^{3} + 2620x^{2} + 4772x - 632$

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
$\operatorname{tr} a_p$	10	−10	−28	−37	−65	−161	−145	−151	−251	0	23	176	−238	227	−474	−823	521	−448	445	1546	−1310	−218	1261	797	−1230

$\operatorname{tr} a_n$ 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)[0]   # 218.4.a.a

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

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