Atlas › Modular forms › Weight 10 › Level 38 › 38.10.a › 38.10.a.a

Newform orbit 38.10.a.a

$q$ -expansion

$q + 16q^{2} - 119q^{3} + 256q^{4} - 684q^{5} - 1904q^{6} + 9149q^{7} + 4096q^{8} - 5522q^{9} - 10944q^{10} + 5790q^{11} - 30464q^{12} - 179881q^{13} + 146384q^{14} + 81396q^{15} + 65536q^{16} - 594093q^{17} - 88352q^{18} + 130321q^{19} - 175104q^{20} - 1088731q^{21} + 92640q^{22} - 1744767q^{23} - 487424q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{5}$ is a root of

$x + 803$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
19	$-1$

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
$\operatorname{tr} a_p$	16	−119	−684	9149	5790	−179881	−594093	130321	−1744767	4314387	160232	−21943090	294816	−39393148	46596360	22121703	33070233	188535938	−20769067	−232299978	−3022183	−446379406	794022846	90999336	−123974170

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

Sato–Tate

The normalized $a_p / 2p^{9/2}$ for the primes $p < 1000$ not dividing 38, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.333 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, 2, 19, 38]: [1, 3, 16, 47] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 5 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(38, 10)[0]   # 38.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 38:10 1000