Atlas › Modular forms › Weight 4 › Level 46 › 46.4.a › 46.4.a.b

Newform orbit 46.4.a.b

$q$ -expansion

$q + 2q^{2} - 9q^{3} + 4q^{4} - 20q^{5} - 18q^{6} + 2q^{7} + 8q^{8} + 54q^{9} - 40q^{10} - 52q^{11} - 36q^{12} + 43q^{13} + 4q^{14} + 180q^{15} + 16q^{16} - 50q^{17} + 108q^{18} - 74q^{19} - 80q^{20} - 18q^{21} - 104q^{22} - 23q^{23} - 72q^{24} + O(q^{25})$

Coefficient field

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

$x + 69$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
23	$+1$

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	−9	−20	2	−52	43	−50	−74	−23	−7	−273	−4	123	−152	75	86	−444	262	764	−21	681	426	902	−1272	−342

$\operatorname{tr} a_n$ 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 46, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.102 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, 23, 46]: [1, 2, 7, 20] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 6 = 1 x 6 (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(46, 4)[1]   # 46.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 46:4 1000