Atlas › Modular forms › Weight 12 › Level 4 › 4.12.a › 4.12.a.a

Newform orbit 4.12.a.a

$q$ -expansion

$q - 516q^{3} - 10530q^{5} + 49304q^{7} + 89109q^{9} - 309420q^{11} - 1723594q^{13} + 5433480q^{15} - 2279502q^{17} + 4550444q^{19} - 25440864q^{21} - 7282872q^{23} + O(q^{25})$

Coefficient field

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

$x + 11046$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 4$ )

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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	−516	−10530	49304	−309420	−1723594	−2279502	4550444	−7282872	−69040026	−141740704	711366974	−1225262214	−33606220	123214608	1106121582	−9062779932	−3854150458	−15313764676	20619626328	−2063718694	13689871472	65570428908	−29715508854	−23439626206

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

Sato–Tate

The normalized $a_p / 2p^{11/2}$ for the primes $p < 1000$ not dividing 4, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.292 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, 4]: [2, 4, 7] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 1 = 1 x 1 (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(4, 12)[0]   # 4.12.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 4:12 1000