Atlas › Modular forms › Weight 10 › Level 18 › 18.10.a › 18.10.a.b

Newform orbit 18.10.a.b

$q$ -expansion

$q - 16q^{2} + 256q^{4} - 384q^{5} + 5852q^{7} - 4096q^{8} + 6144q^{10} - 90624q^{11} - 102814q^{13} - 93632q^{14} + 65536q^{16} - 458496q^{17} - 128824q^{19} - 98304q^{20} + 1449984q^{22} + 1274880q^{23} + O(q^{25})$

Coefficient field

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

$x - 5468$

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
$\operatorname{tr} a_p$	−16	0	−384	5852	−90624	−102814	−458496	−128824	1274880	4884864	−7727524	3121238	−25186560	10223048	−19430400	59935104	75334656	207606062	−178167184	−4902912	−42043210	−364859044	−317941248	788009472	631569422

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

Sato–Tate

The normalized $a_p / 2p^{9/2}$ 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 0.311 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]: [1, 3, 4, 11, 10, 29] (certified at ell = 2147483647)
T = [(5, 1), (7, 1)]; h of degree 4 = 1 x 4 (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(18, 10)[1]   # 18.10.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 18:10 1000