Atlas › Modular forms › Weight 2 › Level 985 › 985.2.a › 985.2.a.g

Newform orbit 985.2.a.g

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$17q + 6q^{2} + 5q^{3} + 18q^{4} - 17q^{5} + 3q^{6} + 7q^{7} + 18q^{8} + 22q^{9} - 6q^{10} + 7q^{11} + 20q^{12} + 3q^{13} + 17q^{14} - 5q^{15} + 28q^{16} + 6q^{17} + 13q^{18} - 23q^{19} - 18q^{20} - q^{21} + 14q^{22} + 49q^{23} + q^{24} + O(q^{25})$

Coefficient field

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

$x^{17} - 11x^{16} - 5x^{15} + 404x^{14} - 710x^{13} - 5321x^{12} + 13196x^{11} + 31561x^{10} - 85179x^{9} - 95919x^{8} + 220428x^{7} + 166922x^{6} - 199619x^{5} - 105905x^{4} + 62145x^{3} + 20165x^{2} - 5393x + 104$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
197	$-1$

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
$\operatorname{tr} a_p$	6	5	−17	7	7	3	6	−23	49	19	−18	10	14	22	51	37	23	−15	24	68	−10	3	29	−11	6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.287 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, 5, 197, 985]: [0, 1, 17, 100] (certified at ell = 2147479489)
T = [(2, 1), (3, 1)]; h of degree 65 = 1 x 65 (phi(ord eps) x dim S_k^new); 7 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(985, 2)[6]   # 985.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 985:2 1000