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

Newform orbit 170.4.a.c

Trace form

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

$2q - 4q^{2} - 6q^{3} + 8q^{4} + 10q^{5} + 12q^{6} - 34q^{7} - 16q^{8} + 50q^{9} - 20q^{10} - 6q^{11} - 24q^{12} + 24q^{13} + 68q^{14} - 30q^{15} + 32q^{16} - 34q^{17} - 100q^{18} + 4q^{19} + 40q^{20} + 16q^{21} + 12q^{22} - 222q^{23} + 48q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{43})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{7}$ , whose minimal polynomial is

$x^{2} + 108x + 2744$

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 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
$\operatorname{tr} a_p$	−4	−6	10	−34	−6	24	−34	4	−222	−152	−234	0	184	−500	176	652	436	−636	−376	202	−2176	346	−348	404	740

$\operatorname{tr} a_n$ 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)[2]   # 170.4.a.c

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

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