Atlas › Modular forms › Weight 4 › Level 217 › 217.4.a › 217.4.a.e

Newform orbit 217.4.a.e

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q + 6q^{2} + 10q^{3} + 54q^{4} + 46q^{5} + 20q^{6} + 91q^{7} + 75q^{8} + 121q^{9} + 106q^{10} - 16q^{11} - 44q^{12} + 42q^{13} + 42q^{14} + 96q^{15} + 130q^{16} + 236q^{17} - 75q^{18} + 166q^{19} + 565q^{20} + 70q^{21} - 33q^{22} + 400q^{23} + 183q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{13} - 16x^{12} - 207x^{11} + 4168x^{10} + 5878x^{9} - 300482x^{8} + 217240x^{7} + 9325847x^{6} - 12143188x^{5} - 133394361x^{4} + 190762892x^{3} + 765926436x^{2} - 1022569344x - 732729088$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$
31	$-1$

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
$\operatorname{tr} a_p$	6	10	46	91	−16	42	236	166	400	128	403	−530	1034	20	1960	626	1214	−988	232	−332	−748	448	−476	2216	1252

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.144 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, 7, 31, 217]: [1, 3, 9, 66] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; h of degree 46 = 1 x 46 (phi(ord eps) x dim S_k^new); 7 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(217, 4)[4]   # 217.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 217:4 1000