Atlas › Modular forms › Weight 2 › Level 860 › 860.2.a › 860.2.a.b

Newform orbit 860.2.a.b

Trace form

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

$3q + 3q^{3} - 3q^{5} + 3q^{7} + 3q^{11} + 6q^{13} - 3q^{15} + 6q^{17} + 6q^{19} + 12q^{21} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{7}$ , whose minimal polynomial is

$x^{3} - 6x^{2} - 9x + 17$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 860$ )
5	$+1$
43	$+1$

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	−3	3	3	6	6	6	0	0	15	9	3	−3	6	6	−15	18	12	0	−3	−3	0	−30	−12

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.277 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, 5, 10, 20, 43, 86, 172, 215, 430, 860]: [0, 1, 2, 1, 3, 6, 4, 13, 25, 24, 70, 138] (certified at ell = 2147483353)
T = [(3, 1), (7, 1)]; h of degree 14 = 1 x 14 (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(860, 2)[1]   # 860.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 860:2 1000