Atlas › Modular forms › Weight 6 › Level 48 › 48.6.a › 48.6.a.c

Newform orbit 48.6.a.c

$q$ -expansion

$q + 9q^{3} - 66q^{5} - 176q^{7} + 81q^{9} + 60q^{11} - 658q^{13} - 594q^{15} - 414q^{17} - 956q^{19} - 1584q^{21} - 600q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{7}$ is a root of

$x + 594$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 48$ )
3	$-1$

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
$\operatorname{tr} a_p$	0	9	−66	−176	60	−658	−414	−956	−600	5574	3592	−8458	19194	−13316	19680	−31266	−26340	−31090	16804	−6120	−25558	−74408	6468	−32742	166082

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ for the primes $p < 1000$ not dividing 48, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.134 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, 2, 3, 4, 6, 8, 12, 16, 24, 48]: [1, 2, 3, 4, 7, 7, 13, 12, 24, 44] (certified at ell = 2147483629)
T = [(5, 1), (7, 3)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 2 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(48, 6)[2]   # 48.6.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 48:6 1000