Atlas › Modular forms › Weight 8 › Level 13 › 13.8.a › 13.8.a.a

Newform orbit 13.8.a.a

$q$ -expansion

$q + 10q^{2} - 73q^{3} - 28q^{4} - 295q^{5} - 730q^{6} + 1373q^{7} - 1560q^{8} + 3142q^{9} - 2950q^{10} - 7646q^{11} + 2044q^{12} + 2197q^{13} + 13730q^{14} + 21535q^{15} - 12016q^{16} - 4147q^{17} + 31420q^{18} - 3186q^{19} + 8260q^{20} - 100229q^{21} - 76460q^{22} - 17784q^{23} + 113880q^{24} + O(q^{25})$

Coefficient field

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

$x + 63$

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
$\operatorname{tr} a_p$	10	−73	−295	1373	−7646	2197	−4147	−3186	−17784	−93322	−124484	273661	585816	−533559	−530055	−615288	−392514	1878064	−3971438	−3746601	2485802	−1264456	434308	5830810	−2045330

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Sato–Tate

The normalized $a_p / 2p^{7/2}$ 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 0.248 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, 9] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 7 = 1 x 7 (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(13, 8)[0]   # 13.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 13:8 1000