Atlas › Modular forms › Weight 8 › Level 34 › 34.8.a › 34.8.a.e

Newform orbit 34.8.a.e

Trace form

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

$3q + 24q^{2} + 12q^{3} + 192q^{4} + 346q^{5} + 96q^{6} + 1200q^{7} + 1536q^{8} + 1487q^{9} + 2768q^{10} + 8356q^{11} + 768q^{12} + 15762q^{13} + 9600q^{14} + 15688q^{15} + 12288q^{16} + 14739q^{17} + 11896q^{18} + 2404q^{19} + 22144q^{20} - 46112q^{21} + 66848q^{22} - 1344q^{23} + 6144q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} - 358x^{2} - 93748x + 9645976$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
17	$-1$

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
$\operatorname{tr} a_p$	24	12	346	1200	8356	15762	14739	2404	−1344	37938	−286472	−327046	−761234	−1075172	−199760	882970	1364076	5059506	917348	1521984	4677902	−6817544	−5475580	−1842626	8331174

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.277 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, 2, 17, 34]: [1, 3, 12, 34] (certified at ell = 2147483489)
T = [(3, 1), (5, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 3 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(34, 8)[4]   # 34.8.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 34:8 1000