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

Newform orbit 505.2.a.e

Trace form

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

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

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

$x^{9} - 9x^{8} + 7x^{7} + 128x^{6} - 306x^{5} - 216x^{4} + 865x^{3} + 196x^{2} - 723x - 253$

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	7	9	6	3	3	5	−2	19	7	12	6	13	12	34	5	23	−17	17	23	−27	−24	10	−10	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 0.199 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)[4]   # 505.2.a.e

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

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