Atlas › Modular forms › Weight 4 › Level 68 › 68.4.a › 68.4.a.b

Newform orbit 68.4.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 4q^{3} + 26q^{5} + 8q^{7} + 63q^{9} + 60q^{11} + 82q^{13} + 40q^{15} + 51q^{17} - 124q^{19} - 144q^{21} - 120q^{23} + O(q^{25})$

Coefficient field

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

$x^{3} - 30x^{2} + 108x - 88$

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	4	26	8	60	82	51	−124	−120	−46	−272	186	−82	300	−224	202	−612	786	916	−1272	−306	1568	948	478	1318

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.101 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, 3, 6, 16, 30] (certified at ell = 2147483489)
T = [(3, 1), (5, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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, 4)[1]   # 68.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 68:4 1000