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

Newform orbit 169.4.a.k

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q - 5q^{2} + q^{3} + 37q^{4} - 30q^{5} - 48q^{6} - 38q^{7} - 60q^{8} + 66q^{9} - 147q^{10} - 181q^{11} + 39q^{12} - 147q^{14} - 218q^{15} + 269q^{16} - 55q^{17} - 79q^{18} - 161q^{19} - 370q^{20} - 188q^{21} + 340q^{22} - 204q^{23} - 798q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{9} + 2x^{8} - 1299x^{7} + 4757x^{6} + 471879x^{5} - 3349008x^{4} - 44738526x^{3} + 460072563x^{2} - 792625119x - 397062289$

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
$\operatorname{tr} a_p$	−5	1	−30	−38	−181	0	−55	−161	−204	280	−706	−298	−1201	−533	−956	−278	−1377	−136	931	−2046	45	412	−3709	−1663	1087

$\operatorname{tr} a_n$ 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)[10]   # 169.4.a.k

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

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