Atlas › Modular forms › Weight 6 › Level 92 › 92.6.a › 92.6.a.b

Newform orbit 92.6.a.b

Trace form

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

$4q + 29q^{3} + 14q^{5} + 248q^{7} + 327q^{9} - 80q^{11} + 1331q^{13} + 2046q^{15} + 1690q^{17} + 2752q^{19} + 3100q^{21} - 2116q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} - 43x^{3} - 11617x^{2} + 361615x + 27042300$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 92$ )
23	$+1$

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
$\operatorname{tr} a_p$	0	29	14	248	−80	1331	1690	2752	−2116	9281	8105	9834	−11653	22358	2013	−53614	−19660	−43446	−56544	15921	−179843	116422	−9838	−148986	72030

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.256 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, 2, 4, 23, 46, 92]: [1, 2, 4, 11, 32, 63] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 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(92, 6)[1]   # 92.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 92:6 1000