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

Newform orbit 68.6.a.b

Trace form

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

$4q + 10q^{3} - 88q^{5} + 166q^{7} + 804q^{9} + 594q^{11} + 1176q^{13} + 2260q^{15} - 1156q^{17} + 4060q^{19} + 5256q^{21} + 7218q^{23} + O(q^{25})$

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

$x^{4} + 78x^{3} - 9558x^{2} - 512584x + 12125736$

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	10	−88	166	594	1176	−1156	4060	7218	1832	−5274	11072	−25168	−964	−52352	−16592	−28956	33224	−10008	49650	83576	95722	−32292	15808	390144

$\operatorname{tr} a_n$ 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)[1]   # 68.6.a.b

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

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