Atlas › Modular forms › Weight 8 › Level 45 › 45.8.a › 45.8.a.e

Newform orbit 45.8.a.e

$q$ -expansion

$q + 13q^{2} + 41q^{4} + 125q^{5} + 1380q^{7} - 1131q^{8} + 1625q^{10} + 3304q^{11} + 8506q^{13} + 17940q^{14} - 19951q^{16} + 9994q^{17} + 41236q^{19} + 5125q^{20} + 42952q^{22} - 84120q^{23} + O(q^{25})$

Coefficient field

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

$x - 1393$

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
$\operatorname{tr} a_p$	13	0	125	1380	3304	8506	9994	41236	−84120	−132802	−55800	228170	139670	−755492	−836984	−1641650	989656	−1658162	−4523844	389408	5617330	3901080	9394116	−2803746	5099426

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

Sato–Tate

The normalized $a_p / 2p^{7/2}$ 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 0.298 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, 5, 8, 16, 44] (certified at ell = 2147483629)
T = [(2, 1), (7, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 4 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, 8)[4]   # 45.8.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 45:8 1000