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

Newform orbit 169.4.a.h

Trace form

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

$2q - 14q^{3} + 8q^{4} + 44q^{9} + 96q^{10} - 56q^{12} - 156q^{14} - 160q^{16} - 54q^{17} - 156q^{22} - 114q^{23} + O(q^{25})$

Coefficient field

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

$x^{2} + 42x + 429$

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
$\operatorname{tr} a_p$	0	−14	0	0	0	0	−54	0	−114	−138	0	0	0	170	0	852	0	−34	0	0	0	−2488	0	0	0

$\operatorname{tr} a_n$ 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)[7]   # 169.4.a.h

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

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