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

Newform orbit 45.8.a.h

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 20q^{2} + 96q^{4} + 250q^{5} - 100q^{7} - 1440q^{8} - 2500q^{10} - 4544q^{11} + 3540q^{13} - 7512q^{14} + 20352q^{16} + 27340q^{17} + 38760q^{19} + 12000q^{20} + 106240q^{22} + 124140q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{19})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{7}$ , whose minimal polynomial is

$x^{2} + 120x - 226300$

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
$\operatorname{tr} a_p$	−20	0	250	−100	−4544	3540	27340	38760	124140	72260	306824	−123020	−264364	423300	105460	2391580	1120120	2257044	4516460	−621784	4569060	4333040	9793020	−6025620	4609540

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

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)[7]   # 45.8.a.h

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

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