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

Newform orbit 565.2.a.d

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} + 10x^{9} + 15x^{8} - 90x^{7} - 167x^{6} + 292x^{5} + 348x^{4} - 290x^{3} - 138x^{2} + 120x - 19$

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 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
$\operatorname{tr} a_p$	−5	−5	−10	−6	−21	−4	−12	−2	−16	−12	21	−13	−2	−33	−37	−16	−41	40	−30	8	−11	7	−52	−11	−1

$\operatorname{tr} a_n$ 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)[3]   # 565.2.a.d

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

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