Atlas › Modular forms › Weight 6 › Level 65 › 65.6.a › 65.6.a.a

Newform orbit 65.6.a.a

$q$ -expansion

$q + 5q^{2} + 6q^{3} - 7q^{4} - 25q^{5} + 30q^{6} - 244q^{7} - 195q^{8} - 207q^{9} - 125q^{10} + 794q^{11} - 42q^{12} - 169q^{13} - 1220q^{14} - 150q^{15} - 751q^{16} - 1534q^{17} - 1035q^{18} + 2706q^{19} + 175q^{20} - 1464q^{21} + 3970q^{22} - 702q^{23} - 1170q^{24} + O(q^{25})$

Coefficient field

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

$x - 11$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
13	$+1$

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
$\operatorname{tr} a_p$	5	6	−25	−244	794	−169	−1534	2706	−702	−5038	−3634	−7058	−294	7618	−3020	626	−30066	−5806	−12436	4734	−14694	−39804	−41776	7970	−78050

$\operatorname{tr} a_n$ 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 65, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.243 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, 13, 65]: [1, 3, 7, 36] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 20 = 1 x 20 (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(65, 6)[0]   # 65.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 65:6 1000