Atlas › Modular forms › Weight 6 › Level 68 › 68.6.a › 68.6.a.a

Newform orbit 68.6.a.a

Trace form

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

$2q - 8q^{3} - 20q^{5} + 128q^{7} - 38q^{9} - 856q^{11} - 996q^{13} - 752q^{15} + 578q^{17} - 3096q^{19} - 5088q^{21} + 1824q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{13})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 28x - 12$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 68$ )
17	$-1$

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	−8	−20	128	−856	−996	578	−3096	1824	−4900	−2672	−8724	−13292	16664	9120	972	59832	10972	−14104	51744	−62316	−35184	−7832	−79788	−153276

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.245 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, 17, 34, 68]: [1, 2, 4, 8, 24, 48] (certified at ell = 2147483489)
T = [(3, 1), (5, 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(68, 6)[0]   # 68.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 68:6 1000