Atlas › Modular forms › Weight 4 › Level 13 › 13.4.a › 13.4.a.a

Newform orbit 13.4.a.a

$q$ -expansion

$q - 5q^{2} - 7q^{3} + 17q^{4} - 7q^{5} + 35q^{6} - 13q^{7} - 45q^{8} + 22q^{9} + 35q^{10} - 26q^{11} - 119q^{12} + 13q^{13} + 65q^{14} + 49q^{15} + 89q^{16} + 77q^{17} - 110q^{18} - 126q^{19} - 119q^{20} + 91q^{21} + 130q^{22} - 96q^{23} + 315q^{24} + O(q^{25})$

Coefficient field

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

$x + 12$

Atkin–Lehner signs

$p$	$w_p$
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	−7	−7	−13	−26	13	77	−126	−96	−82	196	−131	336	−201	−105	−432	−294	−56	478	9	98	1304	−308	−1190	70

$\operatorname{tr} a_n$ 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 13, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.094 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, 13]: [1, 5] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 1 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(13, 4)[0]   # 13.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 13:4 1000