Atlas › Modular forms › Weight 6 › Level 35 › 35.6.a › 35.6.a.a

Newform orbit 35.6.a.a

$q$ -expansion

$q - 8q^{2} + q^{3} + 32q^{4} + 25q^{5} - 8q^{6} + 49q^{7} - 242q^{9} - 200q^{10} - 453q^{11} + 32q^{12} - 969q^{13} - 392q^{14} + 25q^{15} - 1024q^{16} + 1637q^{17} + 1936q^{18} - 1550q^{19} + 800q^{20} + 49q^{21} + 3624q^{22} - 1654q^{23} + O(q^{25})$

Coefficient field

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

$x + 7$

Atkin–Lehner signs

$p$	$w_p$
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
$\operatorname{tr} a_p$	−8	1	25	49	−453	−969	1637	−1550	−1654	−4985	1192	−11018	−1728	−10814	26237	25936	−4580	−12488	−15848	51792	4846	62765	−23644	−147300	−8343

$\operatorname{tr} a_n$ 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 35, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.228 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, 5, 7, 35]: [1, 3, 5, 22] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 10 = 1 x 10 (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(35, 6)[0]   # 35.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 35:6 1000