Atlas › Modular forms › Weight 2 › Level 867 › 867.2.a › 867.2.a.p

Newform orbit 867.2.a.p

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 3q^{2} + 6q^{3} + 9q^{4} - 3q^{5} + 3q^{6} + 3q^{7} + 12q^{8} + 6q^{9} + 12q^{10} - 9q^{11} + 9q^{12} + 9q^{13} - 6q^{14} - 3q^{15} + 15q^{16} + 3q^{18} + 9q^{19} + 6q^{20} + 3q^{21} - 18q^{22} - 9q^{23} + 12q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{19} + 7T_{23} + 8T_{29}$ , whose minimal polynomial is

$x^{6} + 42x^{5} - 4398x^{4} - 294274x^{3} - 5604357x^{2} - 37088124x - 68681953$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
17	— ( $17^2 \mid 867$ )

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
$\operatorname{tr} a_p$	3	6	−3	3	−9	9	0	9	−9	−6	24	−3	−18	0	24	24	−9	21	−6	−27	18	24	6	0	−33

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.671 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, 3, 17, 51, 289, 867]: [0, 1, 2, 8, 26, 104] (certified at ell = 2147483249)
T = [(2, 1), (5, 2), (7, 3), (11, 4), (13, 5), (19, 6), (23, 7), (29, 8)]; h of degree 45 = 1 x 45 (phi(ord eps) x dim S_k^new); 13 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(867, 2)[15]   # 867.2.a.p

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

cargo run --release -p sagebrush-web --example atlas -- 867:2 1000