Atlas › Modular forms › Weight 2 › Level 261 › 261.2.a › 261.2.a.e

Newform orbit 261.2.a.e

Trace form

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

$3q - 2q^{2} + 6q^{4} + 4q^{7} - 3q^{8} - 6q^{10} + 8q^{11} + 4q^{13} + 5q^{14} - 4q^{16} - 4q^{17} - 2q^{19} + 16q^{20} - 13q^{22} - 6q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{3} + 2x^{2} - 14x + 13$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 261$ )
29	$+1$

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	0	0	4	8	4	−4	−2	−6	−3	6	8	2	−4	12	−8	20	4	0	14	−8	−2	8	8	4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.184 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, 3, 9, 29, 87, 261]: [0, 1, 2, 3, 12, 32] (certified at ell = 2147483353)
T = [(2, 1), (5, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 2 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(261, 2)[4]   # 261.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 261:2 1000