Atlas › Modular forms › Weight 4 › Level 170 › 170.4.a › 170.4.a.f

Newform orbit 170.4.a.f

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 4q^{2} + 2q^{3} + 8q^{4} + 10q^{5} + 4q^{6} + 46q^{7} + 16q^{8} - 14q^{9} + 20q^{10} - 14q^{11} + 8q^{12} - 32q^{13} + 92q^{14} + 10q^{15} + 32q^{16} - 34q^{17} - 28q^{18} + 260q^{19} + 40q^{20} + 160q^{21} - 28q^{22} + 2q^{23} + 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{19})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{7}$ , whose minimal polynomial is

$x^{2} - 140x + 3000$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	$-1$
17	$+1$

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
$\operatorname{tr} a_p$	4	2	10	46	−14	−32	−34	260	2	128	−74	88	−448	−36	−176	−212	−1068	−1308	−760	74	1144	234	644	1876	244

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.156 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 engine's checks

dim M^+ at levels [1, 2, 5, 10, 17, 34, 85, 170]: [1, 2, 3, 7, 6, 16, 30, 86] (certified at ell = 2147483489)
T = [(3, 1), (7, 3)]; h of degree 16 = 1 x 16 (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(170, 4)[5]   # 170.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 170:4 1000