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

Newform orbit 217.4.a.d

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q + 7q^{2} + 12q^{3} + 61q^{4} - 14q^{5} - 16q^{6} - 84q^{7} + 60q^{8} + 240q^{9} + 10q^{10} + 92q^{11} + 284q^{12} + 26q^{13} - 49q^{14} + 252q^{15} + 473q^{16} + 66q^{17} + 562q^{18} + 72q^{19} - 35q^{20} - 84q^{21} - q^{22} + 520q^{23} + 219q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{12} - 19x^{11} - 164x^{10} + 3874x^{9} + 9934x^{8} - 279902x^{7} - 357780x^{6} + 8524961x^{5} + 8689299x^{4} - 97236644x^{3} - 57215508x^{2} + 369366624x + 26332800$

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
$\operatorname{tr} a_p$	7	12	−14	−84	92	26	66	72	520	206	−372	1204	−628	484	928	1128	492	−1320	636	2724	−242	624	1270	−4198	−2482

$\operatorname{tr} a_n$ 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)[3]   # 217.4.a.d

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

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