Atlas › Modular forms › Weight 4 › Level 65 › 65.4.a › 65.4.a.a

Newform orbit 65.4.a.a

$q$ -expansion

$q + 5q^{2} + 2q^{3} + 17q^{4} - 5q^{5} + 10q^{6} - 12q^{7} + 45q^{8} - 23q^{9} - 25q^{10} + 14q^{11} + 34q^{12} - 13q^{13} - 60q^{14} - 10q^{15} + 89q^{16} + 98q^{17} - 115q^{18} - 26q^{19} - 85q^{20} - 24q^{21} + 70q^{22} - 114q^{23} + 90q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{3}$ 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 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
$\operatorname{tr} a_p$	5	2	−5	−12	14	−13	98	−26	−114	58	306	86	−374	−314	620	362	266	634	612	−686	202	−516	48	−1230	350

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.103 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, 5, 24] (certified at ell = 2147483629)
T = [(2, 1), (3, 3)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 3 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, 4)[0]   # 65.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 65:4 1000