Atlas › Modular forms › Weight 2 › Level 518 › 518.2.a › 518.2.a.f

Newform orbit 518.2.a.f

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 5q^{2} + q^{3} + 5q^{4} - 3q^{5} + q^{6} + 5q^{7} + 5q^{8} + 8q^{9} - 3q^{10} + 5q^{11} + q^{12} + 5q^{13} + 5q^{14} + 6q^{15} + 5q^{16} - 6q^{17} + 8q^{18} + 8q^{19} - 3q^{20} + q^{21} + 5q^{22} + 5q^{23} + q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} + 8x^{4} - 173x^{3} - 420x^{2} + 6744x - 6388$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
7	$-1$
37	$-1$

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	1	−3	5	5	5	−6	8	5	−1	−7	5	−13	10	6	−18	−10	−1	13	8	−17	1	−8	−6	12

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.229 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, 2, 7, 14, 37, 74, 259, 518]: [0, 1, 1, 4, 3, 11, 26, 80] (certified at ell = 2147483629)
T = [(3, 1), (5, 3)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 3 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(518, 2)[5]   # 518.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 518:2 1000