Atlas › Modular forms › Weight 2 › Level 487 › 487.2.a › 487.2.a.d

Newform orbit 487.2.a.d

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$16q + 7q^{2} - 4q^{3} + 27q^{4} + 9q^{5} + 2q^{6} + q^{7} + 27q^{8} + 22q^{9} + 5q^{10} + q^{11} - 4q^{12} + 4q^{13} - 3q^{14} - 20q^{15} + q^{16} + 38q^{17} + 7q^{19} + 36q^{20} - 31q^{21} + 10q^{22} + 8q^{23} + 7q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} - 3x^{15} - 62x^{14} + 177x^{13} + 1385x^{12} - 3618x^{11} - 14468x^{10} + 33543x^{9} + 74563x^{8} - 149223x^{7} - 176055x^{6} + 294140x^{5} + 154217x^{4} - 174418x^{3} - 58369x^{2} + 18832x + 885$

Atkin–Lehner signs

$p$	$w_p$
487	$-1$

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
$\operatorname{tr} a_p$	7	−4	9	1	1	4	38	7	8	6	−10	−8	68	−21	20	−12	−3	−19	10	26	18	−47	29	46	58

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.196 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, 487]: [0, 41] (certified at ell = 2147481883)
T = [(2, 1), (3, 1)]; h of degree 40 = 1 x 40 (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(487, 2)[3]   # 487.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 487:2 1000