Atlas › Modular forms › Weight 4 › Level 81 › 81.4.a › 81.4.a.b

Newform orbit 81.4.a.b

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 3q^{2} + 17q^{4} + 12q^{5} + 10q^{7} - 87q^{8} + 39q^{10} + 42q^{11} + 64q^{13} + 156q^{14} + 137q^{16} - 18q^{17} + 130q^{19} - 69q^{20} - 120q^{22} - 114q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{57})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{2} - 33x - 426$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 81$ )

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	12	10	42	64	−18	130	−114	240	−8	184	−672	370	276	−324	−204	208	802	−126	−374	1378	−1164	−954	−872

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.107 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 engine's checks

dim M^+ at levels [1, 3, 9, 27, 81]: [1, 2, 4, 10, 28] (certified at ell = 2147483179)
T = [(2, 1), (5, 3)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 3 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(81, 4)[1]   # 81.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 81:4 1000