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

Newform orbit 33.8.a.b

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 19q^{2} + 54q^{3} + 13q^{4} - 34q^{5} - 513q^{6} - 166q^{7} + 627q^{8} + 1458q^{9} + 3686q^{10} - 2662q^{11} + 351q^{12} - 12670q^{13} - 12052q^{14} - 918q^{15} - 2399q^{16} - 62344q^{17} - 13851q^{18} - 37980q^{19} - 64118q^{20} - 4482q^{21} + 25289q^{22} - 90686q^{23} + 16929q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{177})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} + 53x - 66602$

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
$\operatorname{tr} a_p$	−19	54	−34	−166	−2662	−12670	−62344	−37980	−90686	13992	245000	327852	−275932	−244104	536926	−1821882	2502028	−2191098	1674784	−368310	−3336604	−1682618	−8376504	9027204	−16703552

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

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)[1]   # 33.8.a.b

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

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