Atlas › Modular forms › Weight 8 › Level 32 › 32.8.a › 32.8.a.d

Newform orbit 32.8.a.d

Trace form

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

$2q + 16q^{3} - 180q^{5} - 1248q^{7} + 874q^{9} - 9040q^{11} - 2500q^{13} - 42400q^{15} + 17220q^{17} - 74160q^{19} - 81664q^{21} - 19104q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{10})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 164x - 118716$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 32$ )

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	16	−180	−1248	−9040	−2500	17220	−74160	−19104	−245028	251520	−530740	726900	−44496	−494912	1319340	−2618000	836700	−2374128	2836000	−171180	2498880	9531984	7318068	−9316060

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.274 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, 8, 16, 32]: [1, 3, 5, 9, 16, 30] (certified at ell = 2147483497)
T = [(3, 1), (5, 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(32, 8)[3]   # 32.8.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 32:8 1000