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

Newform orbit 878.2.a.f

Trace form

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

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

Coefficient field

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

$x^{3} + 5x^{2} - 36x - 167$

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 odd 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
$\operatorname{tr} a_p$	3	−5	0	−4	−3	−14	6	2	−10	−19	−10	−17	−10	−3	−3	6	−9	14	−10	−25	−12	−9	35	31	27

$\operatorname{tr} a_n$ 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)[5]   # 878.2.a.f

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

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