Atlas › Modular forms › Weight 2 › Level 869 › 869.2.a › 869.2.a.f

Newform orbit 869.2.a.f

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q + q^{2} - 9q^{3} + 7q^{4} - 8q^{5} - 3q^{6} - 3q^{7} + 3q^{8} + 10q^{9} - 8q^{10} - 11q^{11} - 18q^{12} - 13q^{13} - 2q^{14} - 7q^{15} - q^{16} + 5q^{17} - 9q^{18} - 31q^{19} - 24q^{20} + 8q^{21} - q^{22} + 3q^{23} + 6q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{11} + 26x^{10} + 139x^{9} - 1545x^{8} - 16015x^{7} + 9303x^{6} + 451503x^{5} + 553868x^{4} - 4137139x^{3} - 3801413x^{2} + 15060526x - 3353129$

Atkin–Lehner signs

$p$	$w_p$
11	$+1$
79	$+1$

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
$\operatorname{tr} a_p$	1	−9	−8	−3	−11	−13	5	−31	3	11	−13	−16	3	−22	−11	−11	−51	−33	−26	−15	0	−11	−11	2	−10

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.276 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, 11, 79, 869]: [0, 2, 7, 82] (certified at ell = 2147481571)
T = [(2, 1), (3, 3)]; h of degree 65 = 1 x 65 (phi(ord eps) x dim S_k^new); 9 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(869, 2)[5]   # 869.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 869:2 1000