Atlas › Modular forms › Weight 6 › Level 84 › 84.6.a › 84.6.a.d

Newform orbit 84.6.a.d

Trace form

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

$2q + 18q^{3} + 78q^{5} + 98q^{7} + 162q^{9} + 174q^{11} + 208q^{13} + 702q^{15} + 1482q^{17} + 352q^{19} + 882q^{21} + 3354q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{505})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + T_{11}$ , whose minimal polynomial is

$x^{2} - 252x - 56844$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 84$ )
3	$-1$
7	$-1$

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	18	78	98	174	208	1482	352	3354	276	6520	13864	−12930	12712	−28116	−46992	−65556	−13148	−75236	−66042	60496	−34916	−82488	42510	213256

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.302 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, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84]: [1, 2, 3, 4, 7, 5, 13, 12, 16, 23, 44, 86] (certified at ell = 2147483647)
T = [(5, 1), (11, 1)]; h of degree 6 = 1 x 6 (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(84, 6)[3]   # 84.6.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 84:6 1000