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

Newform orbit 217.4.a.b

Trace form

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

$10q - 5q^{2} - 12q^{3} + 37q^{4} - 38q^{5} - 48q^{6} + 70q^{7} - 84q^{8} + 82q^{9} + 30q^{10} - 174q^{11} - 70q^{12} + 36q^{13} - 35q^{14} - 172q^{15} + 105q^{16} - 488q^{17} + 158q^{18} + 112q^{19} - 287q^{20} - 84q^{21} - 187q^{22} - 550q^{23} - 417q^{24} + O(q^{25})$

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

$x^{10} + 17x^{9} - 42x^{8} - 1762x^{7} - 2668x^{6} + 52354x^{5} + 76718x^{4} - 719145x^{3} - 197825x^{2} + 4013190x - 3951342$

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	−12	−38	70	−174	36	−488	112	−550	−366	−310	−68	−620	638	−1444	−1972	−1212	−582	−532	628	824	1750	−3546	−1816	−694

$\operatorname{tr} a_n$ 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)[1]   # 217.4.a.b

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

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