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

Newform orbit 45.8.a.i

Trace form

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

$2q - 7q^{2} + 69q^{4} - 250q^{5} + 1304q^{7} - 1449q^{8} + 875q^{10} - 3448q^{11} - 8988q^{13} + 12264q^{14} - 24495q^{16} + 5492q^{17} - 49584q^{19} - 8625q^{20} - 127364q^{22} - 91848q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{601})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{7}$ , whose minimal polynomial is

$x^{2} - 1297x - 67610$

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
$\operatorname{tr} a_p$	−7	0	−250	1304	−3448	−8988	5492	−49584	−91848	−181772	304232	−502316	−631172	353640	467480	568052	−287224	−2514180	−5073832	3748816	−1477212	−4627720	6072936	−16516356	2723428

$\operatorname{tr} a_n$ 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)[8]   # 45.8.a.i

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

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