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

Newform orbit 49.6.a.g

Trace form

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

$4q - 10q^{2} + 10q^{4} - 270q^{8} + 220q^{9} - 1952q^{11} - 4096q^{15} - 1566q^{16} - 5974q^{18} + 3524q^{22} - 7136q^{23} + O(q^{25})$

Coefficient field

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

$x^{4} + 10x^{3} - 5383x^{2} + 21776x + 1858462$

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
$\operatorname{tr} a_p$	−10	0	0	0	−1952	0	0	0	−7136	−3352	0	−9208	0	20448	0	−102920	0	0	−22896	−153824	0	−90688	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 0.23 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)[6]   # 49.6.a.g

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

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