Atlas › Modular forms › Weight 2 › Level 287 › 287.2.a › 287.2.a.f

Newform orbit 287.2.a.f

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - q^{2} - 4q^{3} + 9q^{4} - q^{5} - 4q^{6} - 6q^{7} + 3q^{8} + 14q^{9} + 10q^{10} + 6q^{11} + 9q^{12} + 7q^{13} + q^{14} - 13q^{15} + 7q^{16} + 7q^{17} - 5q^{18} + 2q^{19} + 11q^{20} + 4q^{21} + 15q^{22} + 20q^{23} - 36q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{6} + 13x^{5} - 58x^{4} - 1219x^{3} - 1360x^{2} + 27138x + 78485$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
41	$-1$

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
$\operatorname{tr} a_p$	−1	−4	−1	−6	6	7	7	2	20	−9	−27	19	6	19	−19	5	−7	−12	27	−6	52	0	12	−38	8

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.177 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, 7, 41, 287]: [0, 1, 4, 30] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; 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(287, 2)[5]   # 287.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 287:2 1000