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

Newform orbit 715.2.a.h

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{8} - 2x^{7} - 178x^{6} + 534x^{5} + 7660x^{4} - 20899x^{3} - 106026x^{2} + 192704x + 409460$

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 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	0	8	2	8	8	5	9	0	8	−12	−3	31	−9	1	12	2	26	−21	10	0	−4	14	30	−7

$\operatorname{tr} a_n$ 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)[7]   # 715.2.a.h

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

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