Atlas › Modular forms › Weight 2 › Level 715 › 715.2.a › 715.2.a.g

Newform orbit 715.2.a.g

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{6} + 13x^{5} - 15x^{4} - 485x^{3} - 301x^{2} + 488x - 112$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
11	$+1$
13	$+1$

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	−4	−6	2	−6	−6	−16	6	−18	−8	4	−4	−4	−4	−10	−42	−14	−2	−12	12	10	4	6	−12	−10

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.365 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, 5, 11, 13, 55, 65, 143, 715]: [0, 1, 2, 1, 8, 8, 16, 88] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 39 = 1 x 39 (phi(ord eps) x dim S_k^new); 6 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(715, 2)[6]   # 715.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 715:2 1000