Atlas › Modular forms › Weight 4 › Level 93 › 93.4.a › 93.4.a.a

Newform orbit 93.4.a.a

$q$ -expansion

$q + 3q^{2} - 3q^{3} + q^{4} - 9q^{5} - 9q^{6} - 34q^{7} - 21q^{8} + 9q^{9} - 27q^{10} + 33q^{11} - 3q^{12} + 65q^{13} - 102q^{14} + 27q^{15} - 71q^{16} - 21q^{17} + 27q^{18} - 97q^{19} - 9q^{20} + 102q^{21} + 99q^{22} - 84q^{23} + 63q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{5}$ is a root of

$x + 6$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
31	$-1$

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
$\operatorname{tr} a_p$	3	−3	−9	−34	33	65	−21	−97	−84	48	31	146	−378	182	−501	−402	102	209	−835	−105	542	1109	−597	−1638	−1483

$\operatorname{tr} a_n$ 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 93, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.108 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, 31, 93]: [1, 2, 9, 34] (certified at ell = 2147482951)
T = [(2, 1), (5, 1)]; h of degree 16 = 1 x 16 (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(93, 4)[0]   # 93.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 93:4 1000