Atlas › Modular forms › Weight 6 › Level 40 › 40.6.a › 40.6.a.d

Newform orbit 40.6.a.d

Trace form

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

$2q - 12q^{3} + 50q^{5} + 52q^{7} + 618q^{9} + 560q^{11} + 1388q^{13} - 300q^{15} + 148q^{17} - 1000q^{19} + 2784q^{21} - 2452q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{129})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{7}$ , whose minimal polynomial is

$x^{2} - 40x - 7856$

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	−12	50	52	560	1388	148	−1000	−2452	1340	−2248	−5940	23076	17684	−2908	−5412	62584	14108	−85412	47208	−67452	−65904	108724	−55020	147668

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.237 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, 2, 4, 3, 7, 9, 18, 34] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 2 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, 6)[3]   # 40.6.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 40:6 1000