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

Newform orbit 45.8.a.c

$q$ -expansion

$q + 5q^{2} - 103q^{4} - 125q^{5} + 930q^{7} - 1155q^{8} - 625q^{10} + 8450q^{11} + 6220q^{13} + 4650q^{14} + 7409q^{16} + 9590q^{17} - 45884q^{19} + 12875q^{20} + 42250q^{22} + 102120q^{23} + O(q^{25})$

Coefficient field

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

$x - 935$

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
$\operatorname{tr} a_p$	5	0	−125	930	8450	6220	9590	−45884	102120	87550	−76212	264440	103600	−324680	−855880	958190	−1239550	628522	310380	3934300	−4556090	5371644	6711060	3346500	15829730

$\operatorname{tr} a_n$ 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)[2]   # 45.8.a.c

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

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