Atlas › Modular forms › Weight 6 › Level 17 › 17.6.a › 17.6.a.a

Newform orbit 17.6.a.a

$q$ -expansion

$q - 6q^{2} + 10q^{3} + 4q^{4} - 72q^{5} - 60q^{6} - 196q^{7} + 168q^{8} - 143q^{9} + 432q^{10} + 450q^{11} + 40q^{12} - 142q^{13} + 1176q^{14} - 720q^{15} - 1136q^{16} - 289q^{17} + 858q^{18} - 244q^{19} - 288q^{20} - 1960q^{21} - 2700q^{22} + 2904q^{23} + 1680q^{24} + O(q^{25})$

Coefficient field

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

$x - 4$

Atkin–Lehner signs

$p$	$w_p$
17	$+1$

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
$\operatorname{tr} a_p$	−6	10	−72	−196	450	−142	−289	−244	2904	−6984	−436	−8572	16374	−19216	−19920	1146	22008	35780	23264	−31704	−13966	−51088	−64344	70650	62702

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ for the primes $p < 1000$ not dividing 17, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.209 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, 17]: [1, 8] (certified at ell = 2147483489)
T = [(2, 1), (3, 1)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 2 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(17, 6)[0]   # 17.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 17:6 1000