Atlas › Modular forms › Weight 8 › Level 33 › 33.8.a › 33.8.a.a

Newform orbit 33.8.a.a

$q$ -expansion

$q + 10q^{2} + 27q^{3} - 28q^{4} - 410q^{5} + 270q^{6} - 1028q^{7} - 1560q^{8} + 729q^{9} - 4100q^{10} - 1331q^{11} - 756q^{12} + 12958q^{13} - 10280q^{14} - 11070q^{15} - 12016q^{16} + 17062q^{17} + 7290q^{18} - 54168q^{19} + 11480q^{20} - 27756q^{21} - 13310q^{22} - 11488q^{23} - 42120q^{24} + O(q^{25})$

Coefficient field

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

$x + 400$

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
$\operatorname{tr} a_p$	10	27	−410	−1028	−1331	12958	17062	−54168	−11488	−186654	−188672	395886	−47546	602088	−647200	−1312722	−2681140	551190	459260	−18072	−426062	297764	5684028	−6342966	16651586

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

Sato–Tate

The normalized $a_p / 2p^{7/2}$ 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 0.274 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, 8, 30] (certified at ell = 2147483171)
T = [(2, 1), (5, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 4 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, 8)[0]   # 33.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 33:8 1000