Atlas › Modular forms › Weight 2 › Level 862 › 862.2.a › 862.2.a.i

Newform orbit 862.2.a.i

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{13}$ , whose minimal polynomial is

$x^{6} + 116x^{5} + 1490x^{4} - 234688x^{3} - 9416921x^{2} - 115487755x - 367227989$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
431	$+1$

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
$\operatorname{tr} a_p$	−6	−2	−2	3	−6	−9	−17	1	−20	1	8	−3	−19	−21	−10	−1	11	−2	4	−4	−39	−16	−20	−17	−29

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.47 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, 2, 431, 862]: [0, 1, 37, 110] (certified at ell = 2147481491)
T = [(3, 1), (5, 3), (7, 5), (11, 7), (13, 9)]; h of degree 35 = 1 x 35 (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(862, 2)[8]   # 862.2.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 862:2 1000