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

Newform orbit 48.6.a.a

$q$ -expansion

$q - 9q^{3} + 6q^{5} + 40q^{7} + 81q^{9} + 564q^{11} + 638q^{13} - 54q^{15} + 882q^{17} + 556q^{19} - 360q^{21} + 840q^{23} + O(q^{25})$

Coefficient field

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

$x - 126$

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	6	40	564	638	882	556	840	4638	−4400	−2410	−6870	−9644	18672	33750	18084	39758	23068	4248	−41110	−21920	−82452	−94086	49442

$\operatorname{tr} a_n$ 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)[0]   # 48.6.a.a

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

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