Atlas › Modular forms › Weight 10 › Level 34 › 34.10.a › 34.10.a.a

Newform orbit 34.10.a.a

Trace form

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

$2q + 32q^{2} + 64q^{3} + 512q^{4} - 2788q^{5} + 1024q^{6} - 728q^{7} + 8192q^{8} - 35942q^{9} - 44608q^{10} - 74368q^{11} + 16384q^{12} - 114980q^{13} - 11648q^{14} - 47936q^{15} + 131072q^{16} + 167042q^{17} - 575072q^{18} - 915016q^{19} - 713728q^{20} - 610848q^{21} - 1189888q^{22} - 912792q^{23} + 262144q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{43})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 2724x + 1193876$

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 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
$\operatorname{tr} a_p$	32	64	−2788	−728	−74368	−114980	167042	−915016	−912792	323436	−3608344	−5274276	7571252	25003976	31608704	−23453812	6401832	178011724	196029000	−92217576	−521865228	−902556040	−182662856	−34540492	−2039669756

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.329 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, 14, 42] (certified at ell = 2147483489)
T = [(3, 1), (5, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 5 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, 10)[0]   # 34.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 34:10 1000