Atlas › Modular forms › Weight 6 › Level 70 › 70.6.a › 70.6.a.f

Newform orbit 70.6.a.f

$q$ -expansion

$q + 4q^{2} - 11q^{3} + 16q^{4} - 25q^{5} - 44q^{6} + 49q^{7} + 64q^{8} - 122q^{9} - 100q^{10} - 267q^{11} - 176q^{12} - 1087q^{13} + 196q^{14} + 275q^{15} + 256q^{16} - 513q^{17} - 488q^{18} - 802q^{19} - 400q^{20} - 539q^{21} - 1068q^{22} - 1290q^{23} - 704q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{11}$ is a root of

$x + 278$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	$+1$
7	$-1$

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
$\operatorname{tr} a_p$	4	−11	−25	49	−267	−1087	−513	−802	−1290	1779	−2584	13862	−11904	−598	−17019	27852	30912	−1780	25052	−51984	47690	−102121	−83676	−32400	−148645

$\operatorname{tr} a_n$ 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 70, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.287 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, 7, 10, 14, 35, 70]: [1, 2, 3, 5, 9, 12, 22, 64] (certified at ell = 2147483629)
T = [(3, 1), (11, 1)]; h of degree 10 = 1 x 10 (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(70, 6)[5]   # 70.6.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 70:6 1000