Atlas › Modular forms › Weight 6 › Level 34 › 34.6.a › 34.6.a.a

Newform orbit 34.6.a.a

$q$ -expansion

$q + 4q^{2} - 14q^{3} + 16q^{4} - 74q^{5} - 56q^{6} - 78q^{7} + 64q^{8} - 47q^{9} - 296q^{10} - 250q^{11} - 224q^{12} + 130q^{13} - 312q^{14} + 1036q^{15} + 256q^{16} + 289q^{17} - 188q^{18} + 424q^{19} - 1184q^{20} + 1092q^{21} - 1000q^{22} - 1734q^{23} - 896q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{5}$ is a root of

$x + 88$

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
$\operatorname{tr} a_p$	4	−14	−74	−78	−250	130	289	424	−1734	−3394	8414	−13634	−19766	−19968	14016	2166	−12672	−11002	34644	−32322	61186	106618	−33920	40086	105058

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ for the primes $p < 1000$ not dividing 34, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.229 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, 2, 8, 24] (certified at ell = 2147483489)
T = [(3, 1), (5, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 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, 6)[0]   # 34.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 34:6 1000