Atlas › Modular forms › Weight 12 › Level 18 › 18.12.a › 18.12.a.a

Newform orbit 18.12.a.a

$q$ -expansion

$q - 32q^{2} + 1024q^{4} - 3630q^{5} + 32936q^{7} - 32768q^{8} + 116160q^{10} + 758748q^{11} - 2482858q^{13} - 1053952q^{14} + 1048576q^{16} - 8290386q^{17} - 10867300q^{19} - 3717120q^{20} - 24279936q^{22} - 20539272q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + T_{7}$ is a root of

$x - 29306$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	— ( $3^2 \mid 18$ )

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
$\operatorname{tr} a_p$	−32	0	−3630	32936	758748	−2482858	−8290386	−10867300	−20539272	−28814550	150501392	−319891714	368008998	620469572	−2763110256	268284258	−1672894740	−7787197498	18706694156	8346990888	19641746522	−5873807200	−8492558172	−75527864010	−82356782494

$\operatorname{tr} a_n$ 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 18, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.361 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, 3, 6, 9, 18]: [2, 4, 5, 13, 12, 35] (certified at ell = 2147483647)
T = [(5, 1), (7, 1)]; h of degree 5 = 1 x 5 (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(18, 12)[0]   # 18.12.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 18:12 1000