Atlas › Modular forms › Weight 2 › Level 981 › 981.2.a › 981.2.a.h

Newform orbit 981.2.a.h

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{9} + 6x^{8} - 284x^{7} - 1448x^{6} + 26254x^{5} + 109802x^{4} - 789947x^{3} - 2630896x^{2} + 1184976x + 2803226$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 981$ )
109	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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
$\operatorname{tr} a_p$	−3	0	−1	6	5	6	0	13	5	3	18	16	−2	4	−3	0	22	−3	14	26	17	−22	14	5	27

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.305 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, 3, 9, 109, 327, 981]: [0, 1, 2, 9, 38, 112] (certified at ell = 2147482801)
T = [(2, 1), (5, 3)]; h of degree 45 = 1 x 45 (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(981, 2)[7]   # 981.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 981:2 1000