Atlas › Modular forms › Weight 2 › Level 514 › 514.2.a › 514.2.a.f

Newform orbit 514.2.a.f

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{9} - 6x^{8} - 38x^{7} + 288x^{6} - 94x^{5} - 2726x^{4} + 7204x^{3} - 7472x^{2} + 3380x - 548$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
257	$+1$

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
$\operatorname{tr} a_p$	9	2	4	−4	12	6	10	2	−4	10	−4	8	−4	−4	−18	4	4	−6	−4	28	−6	−20	−8	34	−4

$\operatorname{tr} a_n$ 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, 2, 257, 514]: [0, 1, 22, 66] (certified at ell = 2147483137)
T = [(3, 1), (5, 1)]; h of degree 21 = 1 x 21 (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(514, 2)[5]   # 514.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 514:2 1000