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

Newform orbit 679.2.a.f

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$17q + 4q^{2} + 22q^{4} + 5q^{5} - 3q^{6} - 17q^{7} + 12q^{8} + 39q^{9} + 4q^{10} + 17q^{11} - 5q^{12} + q^{13} - 4q^{14} + 14q^{15} + 24q^{16} + 13q^{17} + 14q^{18} + 10q^{20} + 11q^{22} + 15q^{23} + 4q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 17. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{17} - 4x^{16} - 416x^{15} + 1511x^{14} + 69929x^{13} - 207648x^{12} - 6252306x^{11} + 13046898x^{10} + 324412884x^{9} - 345709103x^{8} - 9852568860x^{7} + 120765172x^{6} + 164319593677x^{5} + 156851707619x^{4} - 1223922888524x^{3} - 2139282760156x^{2} + 1564661463040x + 3477273155536$

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 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
$\operatorname{tr} a_p$	4	0	5	−17	17	1	13	0	15	25	−5	13	22	1	−28	36	0	22	−15	16	−7	27	−29	73	17

$\operatorname{tr} a_n$ 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)[5]   # 679.2.a.f

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

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