Atlas › Modular forms › Weight 2 › Level 868 › 868.2.a › 868.2.a.d

Newform orbit 868.2.a.d

Trace form

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

$3q + 3q^{3} + 2q^{5} - 3q^{7} + 16q^{9} + 2q^{11} + 7q^{13} - 3q^{15} - 7q^{19} - 3q^{21} + O(q^{25})$

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

$x^{3} - 5x^{2} - 4x + 11$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 868$ )
7	$+1$
31	$+1$

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	3	2	−3	2	7	0	−7	0	3	−3	10	9	15	−15	19	−7	−3	4	11	−15	−20	13	17	−5

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.276 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 4, 7, 14, 28, 31, 62, 124, 217, 434, 868]: [0, 1, 2, 1, 4, 7, 3, 10, 19, 22, 68, 134] (certified at ell = 2147482951)
T = [(3, 1), (5, 1)]; h of degree 16 = 1 x 16 (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(868, 2)[3]   # 868.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 868:2 1000