Atlas › Modular forms › Weight 2 › Level 997 › 997.2.a › 997.2.a.f

Newform orbit 997.2.a.f

Trace form

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

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

Coefficient field

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

$x^{5} + 104x^{4} + 96x^{3} - 194064x^{2} - 4105620x - 22572344$

Atkin–Lehner signs

$p$	$w_p$
997	$+1$

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 odd 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
$\operatorname{tr} a_p$	0	−1	−10	−10	−6	−1	6	4	−15	−4	0	0	−12	−4	6	1	9	40	−5	11	−10	−7	−16	5	−26

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.719 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, 997]: [0, 83] (certified at ell = 2147475601)
T = [(2, 1), (3, 3), (7, 5), (11, 7), (13, 9)]; h of degree 82 = 1 x 82 (phi(ord eps) x dim S_k^new); 20 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(997, 2)[5]   # 997.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 997:2 1000