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

Newform orbit 33.8.a.e

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 15q^{2} + 108q^{3} + 565q^{4} + 306q^{5} + 405q^{6} + 890q^{7} + 2457q^{8} + 2916q^{9} - 8946q^{10} + 5324q^{11} + 15255q^{12} - 1822q^{13} + 35340q^{14} + 8262q^{15} + 65041q^{16} + 32856q^{17} + 10935q^{18} - 12784q^{19} - 4002q^{20} + 24030q^{21} + 19965q^{22} + 114858q^{23} + 66339q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{4} - 321x^{3} - 132750x^{2} + 38510636x + 1253935800$

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
$\operatorname{tr} a_p$	15	108	306	890	5324	−1822	32856	−12784	114858	−104952	−24976	−498856	734556	−201916	1995894	929970	1353156	3998774	1722008	5571858	5600528	−7710226	3431856	4611528	1401692

$\operatorname{tr} a_n$ 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)[4]   # 33.8.a.e

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

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