Atlas › Modular forms › Weight 6 › Level 33 › 33.6.a › 33.6.a.b

Newform orbit 33.6.a.b

$q$ -expansion

$q + q^{2} + 9q^{3} - 31q^{4} - 92q^{5} + 9q^{6} - 26q^{7} - 63q^{8} + 81q^{9} - 92q^{10} + 121q^{11} - 279q^{12} - 692q^{13} - 26q^{14} - 828q^{15} + 929q^{16} - 1442q^{17} + 81q^{18} + 2160q^{19} + 2852q^{20} - 234q^{21} + 121q^{22} - 1582q^{23} - 567q^{24} + O(q^{25})$

Coefficient field

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

$x + 91$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
11	$-1$

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	9	−92	−26	121	−692	−1442	2160	−1582	−5526	4792	−10194	−10622	8580	−2362	−30804	6416	42096	−28444	45690	−18374	−105214	62292	−72246	79262

$\operatorname{tr} a_n$ 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 33, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.227 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, 11, 33]: [1, 3, 6, 22] (certified at ell = 2147483171)
T = [(2, 1), (5, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 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(33, 6)[1]   # 33.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 33:6 1000