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

Newform orbit 33.8.a.d

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 9q^{2} - 81q^{3} - 15q^{4} - 444q^{5} - 243q^{6} + 1614q^{7} + 3153q^{8} + 2187q^{9} + 2880q^{10} - 3993q^{11} + 405q^{12} + 20772q^{13} + 36258q^{14} + 11988q^{15} + 12225q^{16} - 14538q^{17} + 6561q^{18} + 24492q^{19} + 80112q^{20} - 43578q^{21} - 11979q^{22} + 35094q^{23} - 85131q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{3} + 435x^{2} - 40200x - 1533356$

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 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
$\operatorname{tr} a_p$	9	−81	−444	1614	−3993	20772	−14538	24492	35094	−179862	288888	107562	−135198	193536	−591486	79044	2532768	6678792	7150356	1390398	−6429114	6873186	6505596	−8842962	−1764774

$\operatorname{tr} a_n$ 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)[3]   # 33.8.a.d

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

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