Atlas › Modular forms › Weight 2 › Level 282 › 282.2.a › 282.2.a.e

Newform orbit 282.2.a.e

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7} + 5T_{13} + 7T_{17} + 9T_{19}$ , whose minimal polynomial is

$x^{3} + 48x^{2} - 1932x - 92996$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$-1$
47	$-1$

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	3	2	0	−6	0	−2	−4	−8	−6	6	−2	−12	4	3	−2	−16	22	−8	0	10	12	−8	2	18

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.224 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, 3, 6, 47, 94, 141, 282]: [0, 1, 1, 3, 5, 14, 18, 52] (certified at ell = 2147483137)
T = [(5, 1), (7, 3), (13, 5), (17, 7), (19, 9)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 3 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(282, 2)[4]   # 282.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 282:2 1000