Atlas › Modular forms › Weight 10 › Level 32 › 32.10.a › 32.10.a.b

Newform orbit 32.10.a.b

Trace form

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

$2q - 176q^{3} + 1404q^{5} - 2784q^{7} + 30394q^{9} + 6896q^{11} - 45556q^{13} - 557728q^{15} - 472188q^{17} - 780912q^{19} + 787712q^{21} - 3883680q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{106})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} - 1228x - 952668$

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	−176	1404	−2784	6896	−45556	−472188	−780912	−3883680	3909612	−4857216	8608892	−32359020	32230512	20943040	51108252	−63098960	25048812	−224200368	−6805472	−23217420	−726598848	−27482736	−71429100	1850743460

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.324 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, 6, 11, 20, 38] (certified at ell = 2147483497)
T = [(3, 1), (5, 1)]; h of degree 9 = 1 x 9 (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(32, 10)[1]   # 32.10.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 32:10 1000