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

Newform orbit 561.2.a.i

Trace form

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

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

Coefficient field

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

$x^{3} - 81x^{2} + 1368x - 6534$

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	−3	4	2	−3	−8	3	0	12	−2	24	−16	6	−10	20	−10	4	−6	2	2	−16	20	−6	−2	−16

$\operatorname{tr} a_n$ 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)[8]   # 561.2.a.i

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

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