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

Newform orbit 49.8.a.g

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 30q^{2} + 458q^{4} + 4290q^{8} + 7600q^{9} + 17760q^{11} + 29216q^{15} + 33762q^{16} + 244850q^{18} + 430660q^{22} + 96000q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} - 30x^{7} - 12839x^{6} + 79720x^{5} + 43881102x^{4} + 854201920x^{3} - 16274303296x^{2} - 526621670400x - 3184201775104$

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
$\operatorname{tr} a_p$	30	0	0	0	17760	0	0	0	96000	236280	0	150040	0	−253440	0	−3678240	0	0	2060640	10702176	0	24451744	0	0	0

$\operatorname{tr} a_n$ 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)[6]   # 49.8.a.g

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

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