Atlas › Modular forms › Weight 2 › Level 680 › 680.2.a › 680.2.a.f

Newform orbit 680.2.a.f

Trace form

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

$3q + q^{3} - 3q^{5} + 4q^{9} + 2q^{11} + 5q^{13} - q^{15} + 3q^{17} + 11q^{19} + 10q^{21} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{7}$ , whose minimal polynomial is

$x^{3} - x^{2} - 180x + 208$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 680$ )
5	$+1$
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	1	−3	0	2	5	3	11	0	5	3	4	6	10	−25	19	7	3	−10	−25	9	−2	−16	15	−13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.404 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, 340, 680]: [0, 1, 2, 1, 3, 3, 2, 6, 6, 10, 12, 10, 22, 30, 60, 116] (certified at ell = 2147483489)
T = [(3, 1), (7, 3)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 3 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(680, 2)[5]   # 680.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 680:2 1000