Atlas › Modular forms › Weight 2 › Level 863 › 863.2.a › 863.2.a.a

Newform orbit 863.2.a.a

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} + 3x^{3} - 9x^{2} - 17x + 31$

Atkin–Lehner signs

$p$	$w_p$
863	$+1$

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	−4	−9	−4	0	2	−2	3	−3	6	−4	−13	26	−3	−1	−12	−4	−21	−19	7	−10	10	−2	−29	−48

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.27 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, 863]: [0, 73] (certified at ell = 2147466121)
T = [(2, 1), (3, 1)]; h of degree 72 = 1 x 72 (phi(ord eps) x dim S_k^new); 7 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(863, 2)[0]   # 863.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 863:2 1000