Atlas › Modular forms › Weight 4 › Level 34 › 34.4.a › 34.4.a.a

Newform orbit 34.4.a.a

$q$ -expansion

$q - 2q^{2} - 2q^{3} + 4q^{4} - 18q^{5} + 4q^{6} - 10q^{7} - 8q^{8} - 23q^{9} + 36q^{10} - 6q^{11} - 8q^{12} + 74q^{13} + 20q^{14} + 36q^{15} + 16q^{16} + 17q^{17} + 46q^{18} - 88q^{19} - 72q^{20} + 20q^{21} + 12q^{22} - 114q^{23} + 16q^{24} + O(q^{25})$

Coefficient field

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

$x + 20$

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
$\operatorname{tr} a_p$	−2	−2	−18	−10	−6	74	17	−88	−114	−90	−310	86	90	368	−384	−258	240	302	−964	−390	722	−898	912	1446	−1438

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ 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 0.101 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, 6, 16] (certified at ell = 2147483489)
T = [(3, 1), (5, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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, 4)[0]   # 34.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 34:4 1000