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

Newform orbit 878.2.a.i

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{7} - 15x^{6} - 131x^{5} + 2992x^{4} - 10711x^{3} - 15230x^{2} + 36742x + 5431$

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
$\operatorname{tr} a_p$	7	3	4	14	1	2	2	14	−2	9	0	7	0	9	−3	−6	−1	22	20	13	−10	25	−9	−19	−7

$\operatorname{tr} a_n$ 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)[8]   # 878.2.a.i

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

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