Atlas › Modular forms › Weight 2 › Level 706 › 706.2.a › 706.2.a.g

Newform orbit 706.2.a.g

Trace form

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

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

Coefficient field

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

$x^{8} - 17x^{7} - 102x^{6} + 2596x^{5} - 4397x^{4} - 90535x^{3} + 475748x^{2} - 768958x + 388126$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
353	$-1$

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	2	5	4	15	−9	3	−6	20	14	11	−19	11	−3	20	20	28	−18	−4	39	−21	22	15	26	−29

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.342 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, 353, 706]: [0, 1, 30, 90] (certified at ell = 2147480897)
T = [(3, 1), (5, 3)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 5 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(706, 2)[6]   # 706.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 706:2 1000