Atlas › Modular forms › Weight 4 › Level 195 › 195.4.a › 195.4.a.a

Newform orbit 195.4.a.a

$q$ -expansion

$q - 5q^{2} + 3q^{3} + 17q^{4} - 5q^{5} - 15q^{6} + 8q^{7} - 45q^{8} + 9q^{9} + 25q^{10} - 56q^{11} + 51q^{12} - 13q^{13} - 40q^{14} - 15q^{15} + 89q^{16} + 58q^{17} - 45q^{18} + 24q^{19} - 85q^{20} + 24q^{21} + 280q^{22} + 36q^{23} - 135q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{7}$ is a root of

$x - 3$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
5	$+1$
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	3	−5	8	−56	−13	58	24	36	−242	−64	−254	−414	−164	−40	82	−744	494	−508	384	462	−816	−92	1210	−530

$\operatorname{tr} a_n$ 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 195, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.161 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, 5, 13, 15, 39, 65, 195]: [1, 2, 3, 5, 8, 16, 24, 88] (certified at ell = 2147483629)
T = [(2, 1), (7, 1)]; h of degree 24 = 1 x 24 (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(195, 4)[0]   # 195.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 195:4 1000