Atlas › Modular forms › Weight 2 › Level 861 › 861.2.a › 861.2.a.f

Newform orbit 861.2.a.f

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{2} - 11x + 26$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
7	$-1$
41	$+1$

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
$\operatorname{tr} a_p$	−1	−2	4	2	9	2	5	6	0	−1	15	7	−2	−13	−5	4	−8	−13	2	13	−9	−2	−10	8	−36

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.417 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, 7, 21, 41, 123, 287, 861]: [0, 1, 1, 4, 4, 16, 30, 116] (certified at ell = 2147482921)
T = [(2, 1), (5, 3)]; h of degree 39 = 1 x 39 (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(861, 2)[5]   # 861.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 861:2 1000