Atlas › Modular forms › Weight 6 › Level 81 › 81.6.a › 81.6.a.c

Newform orbit 81.6.a.c

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 3q^{2} + 49q^{4} - 78q^{5} - 28q^{7} - 375q^{8} + 30q^{10} - 444q^{11} + 182q^{13} - 1392q^{14} + 289q^{16} - 2178q^{17} + 476q^{19} - 6684q^{20} - 1011q^{22} - 8844q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{4} + 81x^{3} - 3915x^{2} - 365229x - 3623130$

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	−78	−28	−444	182	−2178	476	−8844	−12018	−1132	−7588	−1248	6092	60	10476	−2076	−48142	7148	−35928	61226	59516	−117696	−225864	−33976

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.253 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, 3, 6, 16, 46] (certified at ell = 2147483179)
T = [(2, 1), (5, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 5 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, 6)[2]   # 81.6.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 81:6 1000