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

Newform orbit 34.8.a.d

Trace form

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

$2q - 16q^{2} + 16q^{3} + 128q^{4} + 296q^{5} - 128q^{6} - 1136q^{7} - 1024q^{8} + 370q^{9} - 2368q^{10} + 2624q^{11} + 1024q^{12} + 10756q^{13} + 9088q^{14} + 30064q^{15} + 8192q^{16} - 9826q^{17} - 2960q^{18} + 92504q^{19} + 18944q^{20} + 55536q^{21} - 20992q^{22} + 58032q^{23} - 8192q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{577})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} - 312x - 88756$

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
$\operatorname{tr} a_p$	−16	16	296	−1136	2624	10756	−9826	92504	58032	28968	−274384	−410160	579164	212600	464384	1932308	651576	−2369936	−2232984	2962992	−7004844	−240784	−2575352	916724	−22774340

$\operatorname{tr} a_n$ 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)[3]   # 34.8.a.d

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

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