Atlas › Modular forms › Weight 2 › Level 855 › 855.2.a › 855.2.a.j

Newform orbit 855.2.a.j

Trace form

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

$3q - q^{2} + q^{4} + 3q^{5} - 4q^{7} - 3q^{8} - q^{10} - 4q^{11} - 6q^{13} - 10q^{14} - 3q^{16} - 16q^{17} + 3q^{19} + q^{20} + 4q^{22} - 10q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{23} + 7T_{29} + 8T_{31}$ , whose minimal polynomial is

$x^{3} + 107x^{2} - 9717x - 729389$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 855$ )
5	$-1$
19	$-1$

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
$\operatorname{tr} a_p$	−1	0	3	−4	−4	−6	−16	3	−10	2	8	−20	−10	−10	−6	−8	6	6	−2	−10	−4	12	−6	−40	−2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.739 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, 5, 9, 15, 19, 45, 57, 95, 171, 285, 855]: [0, 1, 1, 2, 4, 2, 8, 8, 12, 22, 44, 124] (certified at ell = 2147483629)
T = [(2, 1), (7, 2), (11, 3), (13, 4), (17, 5), (23, 6), (29, 7), (31, 8)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 9 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(855, 2)[9]   # 855.2.a.j

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

cargo run --release -p sagebrush-web --example atlas -- 855:2 1000