Atlas › Modular forms › Weight 2 › Level 505 › 505.2.a › 505.2.a.d

Newform orbit 505.2.a.d

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 2q^{2} + q^{3} + 10q^{4} - 9q^{5} + 4q^{6} + 9q^{8} + 8q^{9} - 2q^{10} + 11q^{11} + 6q^{12} - q^{13} + 16q^{14} - q^{15} + 8q^{16} + 11q^{17} + 11q^{18} - 2q^{19} - 10q^{20} + 2q^{21} - 11q^{22} + 17q^{23} + 28q^{24} + O(q^{25})$

Coefficient field

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

$x^{9} - 3x^{8} - 31x^{7} + 68x^{6} + 314x^{5} - 408x^{4} - 1063x^{3} + 318x^{2} + 157x + 13$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
101	$-1$

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
$\operatorname{tr} a_p$	2	1	−9	0	11	−1	11	−2	17	9	20	−10	7	−32	42	13	39	−1	−47	35	−5	28	26	38	−26

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.199 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, 101, 505]: [0, 1, 9, 52] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 33 = 1 x 33 (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(505, 2)[3]   # 505.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 505:2 1000