Atlas › Modular forms › Weight 4 › Level 105 › 105.4.a › 105.4.a.a

Newform orbit 105.4.a.a

$q$ -expansion

$q - 3q^{3} - 8q^{4} + 5q^{5} + 7q^{7} + 9q^{9} + 42q^{11} + 24q^{12} + 20q^{13} - 15q^{15} + 64q^{16} + 66q^{17} + 38q^{19} - 40q^{20} - 21q^{21} + 12q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{11}$ is a root of

$x - 42$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
5	$-1$
7	$-1$

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
$\operatorname{tr} a_p$	0	−3	5	7	42	20	66	38	12	−258	146	434	−282	20	−72	336	−360	−682	812	810	−124	1136	156	−1038	1208

$\operatorname{tr} a_n$ 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 105, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.124 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, 3, 5, 7, 15, 21, 35, 105]: [1, 2, 3, 3, 8, 10, 14, 52] (certified at ell = 2147483629)
T = [(2, 1), (11, 1)]; h of degree 12 = 1 x 12 (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(105, 4)[0]   # 105.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 105:4 1000