Atlas › Modular forms › Weight 4 › Level 169 › 169.4.a › 169.4.a.g

Newform orbit 169.4.a.g

Trace form

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

$2q - q^{2} + 5q^{3} - 7q^{4} + 3q^{5} + 23q^{6} + 9q^{7} + 3q^{8} + 35q^{9} + 7q^{10} - 80q^{11} - 43q^{12} + 89q^{14} + 33q^{15} - 39q^{16} + 19q^{17} + 110q^{18} + 84q^{19} - 19q^{20} + 303q^{21} + 142q^{22} + 196q^{23} - 273q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{2} - 14x - 376$

Atkin–Lehner signs

$p$	$w_p$
13	— ( $13^2 \mid 169$ )

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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	5	3	9	−80	0	19	84	196	−44	86	−209	230	287	−435	−118	368	−1058	−68	131	−456	−1008	−1958	720	928

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.125 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, 13, 169]: [1, 5, 47] (certified at ell = 2147483053)
T = [(2, 1), (3, 3)]; h of degree 33 = 1 x 33 (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(169, 4)[6]   # 169.4.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 169:4 1000