Atlas › Modular forms › Weight 2 › Level 169 › 169.2.a › 169.2.a.b

Newform orbit 169.2.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 2q^{2} - 2q^{3} - 4q^{5} - q^{6} - 3q^{7} - 3q^{8} - 3q^{9} + 5q^{10} - 8q^{11} - 5q^{14} - 2q^{15} + 2q^{16} - 2q^{17} + 9q^{18} - 4q^{19} + 2q^{21} + 3q^{22} - 5q^{23} + 9q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 4x^{2} + 3x - 1$

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
$\operatorname{tr} a_p$	−2	−2	−4	−3	−8	0	−2	−4	−5	−1	−5	12	−7	13	−18	1	−19	4	−1	−27	9	−5	−7	−11	7

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.165 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 13, 169]: [0, 1, 15] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 1 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, 2)[1]   # 169.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 169:2 1000