Atlas › Modular forms › Weight 8 › Level 52 › 52.8.a › 52.8.a.b

Newform orbit 52.8.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 8q^{5} - 342q^{7} - 879q^{9} - 1790q^{11} - 6591q^{13} - 12888q^{15} - 29760q^{17} - 46722q^{19} - 76086q^{21} - 86752q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{3} - 8x^{2} - 34420x + 1851200$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 52$ )
13	$+1$

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	0	8	−342	−1790	−6591	−29760	−46722	−86752	−205258	−237450	−264408	−23742	317052	909370	476382	2878778	1151886	2887950	2164030	−2303754	877236	−2717442	−13843050	−12574890

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.295 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, 13, 26, 52]: [1, 3, 5, 9, 27, 52] (certified at ell = 2147483629)
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(52, 8)[1]   # 52.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 52:8 1000