Atlas › Modular forms › Weight 4 › Level 170 › 170.4.a › 170.4.a.b

Newform orbit 170.4.a.b

$q$ -expansion

$q + 2q^{2} + 7q^{3} + 4q^{4} + 5q^{5} + 14q^{6} - 10q^{7} + 8q^{8} + 22q^{9} + 10q^{10} + 24q^{11} + 28q^{12} + 41q^{13} - 20q^{14} + 35q^{15} + 16q^{16} - 17q^{17} + 44q^{18} - 103q^{19} + 20q^{20} - 70q^{21} + 48q^{22} - 6q^{23} + 56q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{7}$ is a root of

$x + 23$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	$-1$
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 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
$\operatorname{tr} a_p$	2	7	5	−10	24	41	−17	−103	−6	−45	5	−196	210	−58	−171	3	645	197	−46	−975	−637	272	−72	−609	−847

$\operatorname{tr} a_n$ 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 170, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.156 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, 2, 5, 10, 17, 34, 85, 170]: [1, 2, 3, 7, 6, 16, 30, 86] (certified at ell = 2147483489)
T = [(3, 1), (7, 3)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 4 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(170, 4)[1]   # 170.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 170:4 1000