Atlas › Modular forms › Weight 2 › Level 289 › 289.2.a › 289.2.a.b

Newform orbit 289.2.a.b

Trace form

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

$2q - q^{2} - q^{3} + 3q^{4} - q^{5} - 6q^{6} - 3q^{7} - 6q^{8} + q^{9} + 7q^{10} - 6q^{11} + 5q^{12} - 3q^{13} - 5q^{14} - 6q^{15} - 3q^{16} + 6q^{18} + q^{19} - 8q^{20} + 8q^{21} + 3q^{22} - q^{23} + 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{13})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{2} + 30x + 173$

Atkin–Lehner signs

$p$	$w_p$
17	— ( $17^2 \mid 289$ )

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	−1	−1	−3	−6	−3	0	1	−1	−9	0	6	−12	−12	6	15	12	−4	18	−8	−8	−8	−13	−8	11

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.174 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, 17, 289]: [0, 2, 26] (certified at ell = 2147483249)
T = [(2, 1), (3, 3), (5, 5), (7, 7)]; h of degree 15 = 1 x 15 (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(289, 2)[1]   # 289.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 289:2 1000