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

Newform orbit 985.2.a.e

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} + 5x^{9} - 2x^{8} - 41x^{7} - 35x^{6} + 87x^{5} + 122x^{4} - 12x^{3} - 62x^{2} - 17x + 1$

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
$\operatorname{tr} a_p$	−2	−3	10	−6	−11	−5	−4	−28	−24	−16	−34	5	−15	−5	−14	−15	−49	−16	−2	−64	18	−11	4	−5	20

$\operatorname{tr} a_n$ 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)[4]   # 985.2.a.e

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

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