Atlas › Modular forms › Weight 8 › Level 49 › 49.8.a › 49.8.a.f

Newform orbit 49.8.a.f

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 6q^{2} + 28q^{3} + 348q^{4} + 252q^{5} + 1022q^{6} - 984q^{8} + 2008q^{9} + 4774q^{10} - 3972q^{11} - 5404q^{12} - 1176q^{13} - 16556q^{15} + 57264q^{16} + 56364q^{17} - 35908q^{18} + 41748q^{19} + 162372q^{20} - 152954q^{22} + 131748q^{23} + 389592q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} - 22x^{3} - 6588x^{2} + 96342x + 9600507$

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
$\operatorname{tr} a_p$	−6	28	252	0	−3972	−1176	56364	41748	131748	34056	401212	−5396	410424	46544	1470084	−642372	752220	1325772	−290916	3377760	6706588	3946244	9542064	16165212	1533112

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.28 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, 33] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 5 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, 8)[5]   # 49.8.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 49:8 1000