Atlas › Modular forms › Weight 2 › Level 982 › 982.2.a › 982.2.a.d

Newform orbit 982.2.a.d

Trace form

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

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

Coefficient field

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

$x^{9} - 33x^{8} + 281x^{7} + 1126x^{6} - 23924x^{5} + 56088x^{4} + 342048x^{3} - 1454304x^{2} + 691328x + 1689088$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
491	$-1$

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	0	11	5	4	7	22	−8	12	16	−16	9	4	0	18	17	13	8	0	24	20	−19	58	32	−5

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.304 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, 491, 982]: [0, 1, 42, 125] (certified at ell = 2147474491)
T = [(3, 1), (5, 3)]; h of degree 40 = 1 x 40 (phi(ord eps) x dim S_k^new); 6 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(982, 2)[3]   # 982.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 982:2 1000