Atlas › Modular forms › Weight 4 › Level 205 › 205.4.a › 205.4.a.a

Newform orbit 205.4.a.a

$q$ -expansion

$q - q^{2} + 2q^{3} - 7q^{4} + 5q^{5} - 2q^{6} + 8q^{7} + 15q^{8} - 23q^{9} - 5q^{10} - 54q^{11} - 14q^{12} + 68q^{13} - 8q^{14} + 10q^{15} + 41q^{16} - 10q^{17} + 23q^{18} - 150q^{19} - 35q^{20} + 16q^{21} + 54q^{22} - 64q^{23} + 30q^{24} + O(q^{25})$

Coefficient field

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

$x - 1$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
41	$+1$

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
$\operatorname{tr} a_p$	−1	2	5	8	−54	68	−10	−150	−64	−56	−336	66	−41	188	−536	172	−24	−262	442	652	−54	−104	1236	370	1294

$\operatorname{tr} a_n$ 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 205, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.139 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, 41, 205]: [1, 3, 12, 66] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 40 = 1 x 40 (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(205, 4)[0]   # 205.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 205:4 1000