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

Newform orbit 217.4.a.c

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q - 5q^{2} - 6q^{3} + 37q^{4} - 14q^{5} - 4q^{6} - 70q^{7} - 84q^{8} + 62q^{9} + 10q^{10} - 80q^{11} - 226q^{12} - 98q^{13} + 35q^{14} - 48q^{15} + 9q^{16} + 32q^{17} - 282q^{18} - 72q^{19} - 115q^{20} + 42q^{21} - 221q^{22} - 618q^{23} + 363q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} + 11x^{9} - 160x^{8} - 1250x^{7} + 10078x^{6} + 35662x^{5} - 181704x^{4} - 548381x^{3} + 852035x^{2} + 3501666x + 2491058$

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 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
$\operatorname{tr} a_p$	−5	−6	−14	−70	−80	−98	32	−72	−618	−200	310	−150	−268	−1348	−1576	−666	−364	2024	804	−1036	2256	562	420	−676	−806

$\operatorname{tr} a_n$ 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)[2]   # 217.4.a.c

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

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