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

Newform orbit 865.2.a.b

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q + 3q^{2} + 4q^{3} + 9q^{4} - 13q^{5} + 5q^{6} + 6q^{7} + 9q^{8} + 15q^{9} - 3q^{10} + 2q^{11} + 16q^{12} + 5q^{13} + 13q^{14} - 4q^{15} - 3q^{16} + 12q^{17} + 15q^{18} - 9q^{19} - 9q^{20} + 4q^{21} + 6q^{22} + 34q^{23} + 15q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{13} - 7x^{12} - 25x^{11} + 220x^{10} + 148x^{9} - 2152x^{8} - 248x^{7} + 8370x^{6} + 1169x^{5} - 12570x^{4} - 4681x^{3} + 4658x^{2} + 2524x + 277$

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	−13	6	2	5	12	−9	34	6	5	1	22	12	69	14	48	−13	7	26	−11	−36	44	37	12

$\operatorname{tr} a_n$ 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)[1]   # 865.2.a.b

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

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