Atlas › Modular forms › Weight 6 › Level 49 › 49.6.a › 49.6.a.f

Newform orbit 49.6.a.f

Trace form

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

$2q + 9q^{2} + 6q^{3} + 5q^{4} + 18q^{5} + 198q^{6} - 9q^{8} + 558q^{9} - 204q^{10} + 396q^{11} + 1554q^{12} + 350q^{13} - 1656q^{15} + 113q^{16} - 1800q^{17} + 3537q^{18} + 3266q^{19} - 2520q^{20} - 1752q^{22} + 2088q^{23} + 1854q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{57})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{2} - 27x - 4962$

Atkin–Lehner signs

$p$	$w_p$
7	— ( $7^2 \mid 49$ )

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
$\operatorname{tr} a_p$	9	6	18	0	396	350	−1800	3266	2088	6696	20	6232	6048	−3020	−11700	9468	43938	64754	24784	97416	−17452	51256	−117558	−84276	−20776

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.23 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, 7, 49]: [1, 5, 25] (certified at ell = 2147483647)
T = [(2, 1), (3, 3)]; h of degree 14 = 1 x 14 (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(49, 6)[5]   # 49.6.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 49:6 1000