Atlas › Modular forms › Weight 2 › Level 679 › 679.2.a › 679.2.a.b

Newform orbit 679.2.a.b

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{4} - 2x^{3} - 16x^{2} + 16x + 16$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
97	$+1$

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 odd 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
$\operatorname{tr} a_p$	−1	1	2	−4	−6	−1	−15	−2	−9	−6	−15	−14	−21	9	24	−13	7	3	−18	12	21	12	−17	11	−4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.328 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, 7, 97, 679]: [0, 1, 8, 66] (certified at ell = 2147483137)
T = [(2, 1), (3, 3)]; h of degree 49 = 1 x 49 (phi(ord eps) x dim S_k^new); 7 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(679, 2)[1]   # 679.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 679:2 1000