Atlas › Modular forms › Weight 2 › Level 878 › 878.2.a › 878.2.a.j

Newform orbit 878.2.a.j

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q - 8q^{2} + q^{3} + 8q^{4} - 2q^{5} - q^{6} + 6q^{7} - 8q^{8} + 15q^{9} + 2q^{10} - 13q^{11} + q^{12} + 10q^{13} - 6q^{14} + 5q^{15} + 8q^{16} + 12q^{17} - 15q^{18} + 2q^{19} - 2q^{20} + 4q^{21} + 13q^{22} - 5q^{23} - q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{8} + 5x^{7} - 319x^{6} - 1782x^{5} + 30324x^{4} + 205756x^{3} - 711231x^{2} - 7256761x - 12464981$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
439	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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
$\operatorname{tr} a_p$	−8	1	−2	6	−13	10	12	2	−5	4	25	22	8	16	34	16	−18	−9	28	−7	22	35	22	2	9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.387 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, 439, 878]: [0, 1, 37, 112] (certified at ell = 2147481151)
T = [(3, 1), (5, 3)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 6 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(878, 2)[9]   # 878.2.a.j

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

cargo run --release -p sagebrush-web --example atlas -- 878:2 1000