Atlas › Modular forms › Weight 2 › Level 841 › 841.2.a › 841.2.a.f

Newform orbit 841.2.a.f

Trace form

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

$3q + q^{2} - q^{3} - q^{4} - 3q^{5} + 2q^{6} - 3q^{7} - 4q^{9} + 6q^{10} - 5q^{11} - 2q^{12} + q^{13} - 8q^{14} + 8q^{15} - 5q^{16} - 4q^{17} - 6q^{18} + 3q^{19} + q^{20} + q^{21} - 11q^{22} - 7q^{23} - 7q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 2x^{2} - 29x + 13$

Atkin–Lehner signs

$p$	$w_p$
29	— ( $29^2 \mid 841$ )

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	−3	−3	−5	1	−4	3	−7	0	15	5	−10	−3	19	−9	−28	9	13	−21	−5	15	−9	−7	−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 0.234 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, 29, 841]: [0, 3, 73] (certified at ell = 2147480161)
T = [(2, 1), (3, 3)]; h of degree 54 = 1 x 54 (phi(ord eps) x dim S_k^new); 7 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(841, 2)[5]   # 841.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 841:2 1000