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

Newform orbit 985.2.a.h

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q + 7q^{2} + 3q^{3} + 33q^{4} + 18q^{5} + q^{6} - 8q^{7} + 15q^{8} + 21q^{9} + 7q^{10} + 9q^{11} - 2q^{12} - 4q^{13} + 14q^{14} + 3q^{15} + 43q^{16} - q^{17} + 26q^{18} + 23q^{19} + 33q^{20} + 3q^{21} - 16q^{22} + 48q^{23} - 9q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{18} - 10x^{17} - 23x^{16} + 491x^{15} - 516x^{14} - 7848x^{13} + 15664x^{12} + 54572x^{11} - 122745x^{10} - 207194x^{9} + 394264x^{8} + 479275x^{7} - 479907x^{6} - 564346x^{5} + 118728x^{4} + 201417x^{3} + 25036x^{2} - 6035x + 8$

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
$\operatorname{tr} a_p$	7	3	18	−8	9	−4	−1	23	48	4	51	−18	−5	−19	21	17	27	8	−8	57	−39	34	37	43	−8

$\operatorname{tr} a_n$ 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)[7]   # 985.2.a.h

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

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