Atlas › Modular forms › Weight 6 › Level 45 › 45.6.a › 45.6.a.b

Newform orbit 45.6.a.b

$q$ -expansion

$q - 2q^{2} - 28q^{4} - 25q^{5} + 192q^{7} + 120q^{8} + 50q^{10} + 148q^{11} + 286q^{13} - 384q^{14} + 656q^{16} + 1678q^{17} + 1060q^{19} + 700q^{20} - 296q^{22} - 2976q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{7}$ is a root of

$x - 190$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 45$ )
5	$+1$

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
$\operatorname{tr} a_p$	−2	0	−25	192	148	286	1678	1060	−2976	3410	−2448	182	9398	−1244	12088	−23846	20020	32302	60972	32648	−38774	−33360	−16716	−101370	−119038

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ for the primes $p < 1000$ not dividing 45, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.244 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, 9, 15, 45]: [1, 3, 3, 6, 12, 32] (certified at ell = 2147483629)
T = [(2, 1), (7, 1)]; h of degree 9 = 1 x 9 (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(45, 6)[1]   # 45.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 45:6 1000