Atlas › Modular forms › Weight 2 › Level 865 › 865.2.a › 865.2.a.d

Newform orbit 865.2.a.d

Trace form

The 19 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$19q + 3q^{2} + 4q^{3} + 27q^{4} + 19q^{5} + 7q^{6} + 4q^{7} + 9q^{8} + 23q^{9} + 3q^{10} + 20q^{11} + 2q^{12} - q^{13} + 3q^{14} + 4q^{15} + 39q^{16} + 10q^{17} - 11q^{18} + 17q^{19} + 27q^{20} - 4q^{21} - 24q^{22} + 28q^{23} + 9q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 19. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{19} - 7x^{18} - 55x^{17} + 474x^{16} + 809x^{15} - 11507x^{14} + 1196x^{13} + 128667x^{12} - 100794x^{11} - 750144x^{10} + 773207x^{9} + 2376124x^{8} - 2468501x^{7} - 4000775x^{6} + 3518312x^{5} + 3278448x^{4} - 2089557x^{3} - 1104063x^{2} + 388846x + 57811$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
173	$+1$

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
$\operatorname{tr} a_p$	3	4	19	4	20	−1	10	17	28	0	25	−9	−4	−22	63	6	54	11	−29	46	−9	24	40	19	−38

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.249 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, 5, 173, 865]: [0, 1, 15, 88] (certified at ell = 2147482093)
T = [(2, 1), (3, 1)]; h of degree 57 = 1 x 57 (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(865, 2)[3]   # 865.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 865:2 1000