Atlas › Modular forms › Weight 2 › Level 877 › 877.2.a › 877.2.a.a

Newform orbit 877.2.a.a

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{2} + 6x - 11$

Atkin–Lehner signs

$p$	$w_p$
877	$+1$

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
$\operatorname{tr} a_p$	0	−2	−2	0	1	−2	−2	−3	−10	−15	2	1	−1	6	−9	−1	−6	14	−24	6	−13	0	−33	12	−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 0.295 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, 877]: [0, 73] (certified at ell = 2147478961)
T = [(2, 1), (3, 3)]; h of degree 72 = 1 x 72 (phi(ord eps) x dim S_k^new); 10 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(877, 2)[0]   # 877.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 877:2 1000