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

Newform orbit 218.4.a.d

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 18q^{2} + 11q^{3} + 36q^{4} + 27q^{5} + 22q^{6} + 61q^{7} + 72q^{8} + 126q^{9} + 54q^{10} - 10q^{11} + 44q^{12} + 164q^{13} + 122q^{14} + 76q^{15} + 144q^{16} + 229q^{17} + 252q^{18} + 96q^{19} + 108q^{20} + 181q^{21} - 20q^{22} + 117q^{23} + 88q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{9} - 38x^{8} - 341x^{7} + 29721x^{6} - 281826x^{5} - 2849883x^{4} + 55928450x^{3} - 244510828x^{2} + 166395344x - 30936912$

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 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
$\operatorname{tr} a_p$	18	11	27	61	−10	164	229	96	117	347	267	97	150	68	−26	−176	−1715	−373	−1993	260	754	−2071	−590	381	1308

$\operatorname{tr} a_n$ 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)[3]   # 218.4.a.d

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

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