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

Newform orbit 217.4.a.a

$q$ -expansion

$q - q^{2} - 8q^{3} - 7q^{4} + 4q^{5} + 8q^{6} + 7q^{7} + 15q^{8} + 37q^{9} - 4q^{10} + 66q^{11} + 56q^{12} - 78q^{13} - 7q^{14} - 32q^{15} + 41q^{16} + 78q^{17} - 37q^{18} - 106q^{19} - 28q^{20} - 56q^{21} - 66q^{22} - 28q^{23} - 120q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{3}$ is a root of

$x + 9$

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
$\operatorname{tr} a_p$	−1	−8	4	7	66	−78	78	−106	−28	88	−31	152	−18	−506	−484	364	770	−222	−220	−512	−646	−380	−832	−1402	414

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 217, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

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)[0]   # 217.4.a.a

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

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