Atlas › Modular forms › Weight 2 › Level 987 › 987.2.a › 987.2.a.n

Newform orbit 987.2.a.n

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{9} - 210x^{8} + 9765x^{7} + 767401x^{6} - 90004227x^{5} + 3009418922x^{4} - 30946939269x^{3} - 90411419091x^{2} + 781519717298x - 330431752142$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
7	$-1$
47	$-1$

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
$\operatorname{tr} a_p$	3	9	7	9	11	2	13	−9	12	13	−12	12	7	−9	9	27	5	−21	−2	36	−11	−16	−3	−5	−21

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.658 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, 3, 7, 21, 47, 141, 329, 987]: [0, 1, 1, 4, 5, 18, 34, 132] (certified at ell = 2147483137)
T = [(2, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 12 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(987, 2)[13]   # 987.2.a.n

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

cargo run --release -p sagebrush-web --example atlas -- 987:2 1000