Atlas › Modular forms › Weight 2 › Level 718 › 718.2.a › 718.2.a.g

Newform orbit 718.2.a.g

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{8} - 8x^{7} - 4x^{6} + 183x^{5} - 421x^{4} - 496x^{3} + 2941x^{2} - 3704x + 1520$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
359	$+1$

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
$\operatorname{tr} a_p$	8	4	4	4	5	5	13	8	11	−2	12	−5	−8	0	12	−4	14	−3	−11	0	−19	−6	10	−22	−33

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.336 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, 359, 718]: [0, 1, 31, 92] (certified at ell = 2147476963)
T = [(3, 1), (5, 1)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 4 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(718, 2)[6]   # 718.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 718:2 1000