Atlas › Modular forms › Weight 2 › Level 678 › 678.2.a › 678.2.a.i

Newform orbit 678.2.a.i

Trace form

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

$2q + 2q^{2} + 2q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + q^{7} + 2q^{8} + 2q^{9} + 4q^{10} - 3q^{11} + 2q^{12} - 4q^{13} + q^{14} + 4q^{15} + 2q^{16} + 7q^{17} + 2q^{18} + q^{19} + 4q^{20} + q^{21} - 3q^{22} + q^{23} + 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{61})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{2} - 27x - 1663$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$-1$
113	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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
$\operatorname{tr} a_p$	2	2	4	1	−3	−4	7	1	1	−20	11	−9	−12	−13	13	−1	12	2	3	−1	8	4	−17	−23	−1

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.377 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, 3, 6, 113, 226, 339, 678]: [0, 1, 1, 3, 10, 30, 40, 118] (certified at ell = 2147480161)
T = [(5, 1), (7, 3), (11, 5), (13, 7), (17, 9)]; h of degree 19 = 1 x 19 (phi(ord eps) x dim S_k^new); 5 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(678, 2)[8]   # 678.2.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 678:2 1000