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

Newform orbit 869.2.a.h

Trace form

The 21 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$21q + 7q^{3} + 32q^{4} + 6q^{5} + 4q^{6} + 10q^{7} + 30q^{9} + 3q^{10} - 21q^{11} + 11q^{12} + 10q^{13} + 9q^{14} + 19q^{15} + 50q^{16} - 15q^{17} + q^{18} + 51q^{19} + 13q^{20} + 17q^{21} + 6q^{23} - 7q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 21. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{21} - 21x^{20} - 247x^{19} + 7614x^{18} + 8946x^{17} - 1067704x^{16} + 2144338x^{15} + 77775781x^{14} - 260515540x^{13} - 3297543415x^{12} + 12570628426x^{11} + 86726515131x^{10} - 309241587595x^{9} - 1481669123910x^{8} + 3896096140668x^{7} + 16474383476931x^{6} - 21034782958423x^{5} - 105593824977146x^{4} + 854981931625x^{3} + 277236063945801x^{2} + 266270431842605x + 59634834333366$

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
$\operatorname{tr} a_p$	0	7	6	10	−21	10	−15	51	6	−13	46	15	−41	40	8	3	33	9	33	−3	24	21	47	10	16

$\operatorname{tr} a_n$ 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)[7]   # 869.2.a.h

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

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