Atlas › Modular forms › Weight 4 › Level 201 › 201.4.a › 201.4.a.a

Newform orbit 201.4.a.a

$q$ -expansion

$q - 4q^{2} - 3q^{3} + 8q^{4} - 19q^{5} + 12q^{6} + 13q^{7} + 9q^{9} + 76q^{10} + 26q^{11} - 24q^{12} + 26q^{13} - 52q^{14} + 57q^{15} - 64q^{16} - 96q^{17} - 36q^{18} + 124q^{19} - 152q^{20} - 39q^{21} - 104q^{22} + 153q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{5}$ is a root of

$x + 23$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
67	$-1$

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
$\operatorname{tr} a_p$	−4	−3	−19	13	26	26	−96	124	153	−188	−229	−271	−225	121	272	−503	351	436	67	−792	−97	−848	865	430	−270

$\operatorname{tr} a_n$ 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 201, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.138 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, 3, 67, 201]: [1, 2, 18, 70] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 34 = 1 x 34 (phi(ord eps) x dim S_k^new); 6 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(201, 4)[0]   # 201.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 201:4 1000