Atlas › Modular forms › Weight 6 › Level 95 › 95.6.a › 95.6.a.a

Newform orbit 95.6.a.a

$q$ -expansion

$q - 7q^{2} - 11q^{3} + 17q^{4} - 25q^{5} + 77q^{6} - 197q^{7} + 105q^{8} - 122q^{9} + 175q^{10} - 468q^{11} - 187q^{12} - 921q^{13} + 1379q^{14} + 275q^{15} - 1279q^{16} - 1107q^{17} + 854q^{18} + 361q^{19} - 425q^{20} + 2167q^{21} + 3276q^{22} - 3641q^{23} - 1155q^{24} + O(q^{25})$

Coefficient field

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

$x + 18$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
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 even 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
$\operatorname{tr} a_p$	−7	−11	−25	−197	−468	−921	−1107	361	−3641	7525	1422	−11282	−678	5974	−11072	−17461	−46305	16292	36373	−82208	−43861	−30130	−91626	79170	128718

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ for the primes $p < 1000$ not dividing 95, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.266 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, 19, 95]: [1, 3, 10, 52] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 30 = 1 x 30 (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(95, 6)[0]   # 95.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 95:6 1000