Atlas › Modular forms › Weight 2 › Level 561 › 561.2.a › 561.2.a.k

Newform orbit 561.2.a.k

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{13} + 7T_{19} + 9T_{23}$ , whose minimal polynomial is

$x^{6} - 129x^{5} + 3565x^{4} + 61739x^{3} - 1649240x^{2} - 16418782x + 34462734$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
11	$+1$
17	$+1$

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
$\operatorname{tr} a_p$	−1	6	2	7	−6	8	−6	12	0	−7	14	16	3	8	13	5	13	−16	21	8	3	−4	−6	5	14

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.307 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, 3, 11, 17, 33, 51, 187, 561]: [0, 1, 2, 2, 6, 8, 20, 76] (certified at ell = 2147482801)
T = [(2, 1), (5, 3), (13, 5), (19, 7), (23, 9)]; h of degree 27 = 1 x 27 (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(561, 2)[10]   # 561.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 561:2 1000