Atlas › Modular forms › Weight 10 › Level 40 › 40.10.a › 40.10.a.a

Newform orbit 40.10.a.a

Trace form

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

$2q - 116q^{3} - 1250q^{5} + 11284q^{7} + 37762q^{9} - 101408q^{11} - 21372q^{13} + 72500q^{15} - 296780q^{17} + 275832q^{19} - 3118472q^{21} - 585284q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{22})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{7}$ , whose minimal polynomial is

$x^{2} - 11168x - 9510144$

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	−116	−1250	11284	−101408	−21372	−296780	275832	−585284	−9928756	5131480	−11007932	−41835956	−23394052	11711748	−46384268	178239576	31825220	89480628	112319176	−93294524	−191601328	−17270436	−615067148	−996545468

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.356 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, 6, 5, 11, 15, 30, 58] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 5 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, 10)[0]   # 40.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 40:10 1000