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

Newform orbit 869.2.a.g

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q + 4q^{2} + 8q^{3} + 24q^{4} - 3q^{5} + 2q^{6} + 10q^{7} - 6q^{8} + 18q^{9} + 7q^{10} + 18q^{11} + 5q^{12} + 4q^{13} - 7q^{14} + 4q^{15} + 32q^{16} + 19q^{17} + 9q^{18} + 47q^{19} - 5q^{20} + 7q^{21} + 4q^{22} - q^{23} + 43q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{18} - 28x^{17} + 32x^{16} + 5743x^{15} - 45827x^{14} - 305966x^{13} + 4643000x^{12} - 1448520x^{11} - 177211612x^{10} + 512216945x^{9} + 2794384365x^{8} - 13916499743x^{7} - 12094454138x^{6} + 144153191840x^{5} - 98582750833x^{4} - 523413433970x^{3} + 771016321344x^{2} + 11735199073x - 183397332980$

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 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
$\operatorname{tr} a_p$	4	8	−3	10	18	4	19	47	−1	5	7	−8	37	22	−26	−19	20	47	16	−8	20	−18	15	−21	−17

$\operatorname{tr} a_n$ 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)[6]   # 869.2.a.g

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

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