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

Newform orbit 679.2.a.e

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q + 2q^{2} + 4q^{3} + 16q^{4} + 19q^{5} - q^{6} + 14q^{7} - 3q^{8} + 24q^{9} - 8q^{10} + 5q^{11} + 15q^{12} + 9q^{13} + 2q^{14} - 2q^{15} + 12q^{16} + 29q^{17} - 32q^{18} + 4q^{19} + 24q^{20} + 4q^{21} - 20q^{22} + 17q^{23} - 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{14} - 14x^{13} - 218x^{12} + 3986x^{11} + 5835x^{10} - 338710x^{9} + 938624x^{8} + 9075647x^{7} - 46276642x^{6} - 28249739x^{5} + 405102767x^{4} - 198374080x^{3} - 721098400x^{2} - 189554200x - 7739600$

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
$\operatorname{tr} a_p$	2	4	19	14	5	9	29	4	17	5	9	−15	30	1	36	16	2	2	15	−8	19	−37	37	71	−14

$\operatorname{tr} a_n$ 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)[4]   # 679.2.a.e

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

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