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

Newform orbit 985.2.a.f

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q - 6q^{2} - 7q^{3} + 10q^{4} - 14q^{5} + 5q^{6} - 4q^{7} - 12q^{8} + 13q^{9} + 6q^{10} - 5q^{11} - 10q^{12} - 15q^{13} - 15q^{14} + 7q^{15} + 18q^{16} - 18q^{17} - 29q^{18} + 20q^{19} - 10q^{20} - 5q^{21} + 12q^{22} - 42q^{23} + 7q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} + 13x^{13} + 33x^{12} - 200x^{11} - 918x^{10} + 809x^{9} + 7379x^{8} + 712x^{7} - 25165x^{6} - 8872x^{5} + 36013x^{4} + 11429x^{3} - 20147x^{2} - 4035x + 3551$

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 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
$\operatorname{tr} a_p$	−6	−7	−14	−4	−5	−15	−18	20	−42	−2	0	−15	3	−15	−40	−39	19	−18	−22	−50	−10	1	−12	7	2

$\operatorname{tr} a_n$ 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)[5]   # 985.2.a.f

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

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