Atlas › Modular forms › Weight 2 › Level 565 › 565.2.a › 565.2.a.e

Newform orbit 565.2.a.e

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{12} - 8x^{11} - 16x^{10} + 270x^{9} - 362x^{8} - 1968x^{7} + 4303x^{6} + 3720x^{5} - 10258x^{4} - 3078x^{3} + 7761x^{2} + 1492x - 929$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
113	$+1$

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
$\operatorname{tr} a_p$	5	3	12	0	23	−8	4	20	6	10	17	−11	12	11	−5	10	17	14	0	20	−13	39	32	−3	−47

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.206 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, 5, 113, 565]: [0, 1, 10, 58] (certified at ell = 2147480161)
T = [(2, 1), (3, 1)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 4 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(565, 2)[4]   # 565.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 565:2 1000