Atlas › Modular forms › Weight 8 › Level 40 › 40.8.a › 40.8.a.c

Newform orbit 40.8.a.c

Trace form

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

$2q + 36q^{3} - 250q^{5} - 404q^{7} - 2958q^{9} - 2080q^{11} - 6124q^{13} - 4500q^{15} - 15116q^{17} - 66520q^{19} - 55656q^{21} - 151804q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{6})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{7}$ , whose minimal polynomial is

$x^{2} + 368x - 1442240$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 40$ )
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
$\operatorname{tr} a_p$	0	36	−250	−404	−2080	−6124	−15116	−66520	−151804	16700	−100888	667380	1154796	286868	−212644	−159324	−1589624	1723028	−4790756	2649528	−3722844	−6979536	4161748	−4904940	−15776956

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.294 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, 2, 4, 5, 8, 10, 20, 40]: [1, 3, 5, 5, 9, 13, 24, 46] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 7 = 1 x 7 (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(40, 8)[2]   # 40.8.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 40:8 1000