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

Newform orbit 33.8.a.c

Trace form

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

$2q + q^{2} - 54q^{3} + 181q^{4} - 194q^{5} - 27q^{6} - 418q^{7} + 399q^{8} + 1458q^{9} - 6208q^{10} + 2662q^{11} - 4887q^{12} - 13246q^{13} - 18542q^{14} + 5238q^{15} - 39119q^{16} - 10256q^{17} + 729q^{18} + 14196q^{19} - 23668q^{20} + 11286q^{21} + 1331q^{22} - 13666q^{23} - 10773q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{97})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} + 193x - 27572$

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	−54	−194	−418	2662	−13246	−10256	14196	−13666	15312	−48040	274092	755836	−1704096	−1182094	−2156394	927332	−1061994	−3259952	−5495514	−5450812	−1536590	8850888	6810132	9897376

$\operatorname{tr} a_n$ 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)[2]   # 33.8.a.c

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

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