Atlas › Modular forms › Weight 2 › Level 717 › 717.2.a › 717.2.a.d

Newform orbit 717.2.a.d

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{6} - 3x^{5} - 7x^{4} + 13x^{3} + 2x^{2} - 4x - 1$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
239	$+1$

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
$\operatorname{tr} a_p$	−2	−6	5	−9	−13	−1	11	−22	−12	0	−18	−8	10	−14	−9	−8	−10	−12	−36	−3	−32	−1	−7	17	−28

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.231 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, 3, 239, 717]: [0, 1, 21, 82] (certified at ell = 2147482093)
T = [(2, 1), (5, 1)]; h of degree 39 = 1 x 39 (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(717, 2)[3]   # 717.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 717:2 1000