Atlas › Modular forms › Weight 6 › Level 52 › 52.6.a › 52.6.a.b

Newform orbit 52.6.a.b

$q$ -expansion

$q + 17q^{3} - 91q^{5} - 233q^{7} + 46q^{9} + 134q^{11} + 169q^{13} - 1547q^{15} + 797q^{17} - 1310q^{19} - 3961q^{21} - 2016q^{23} + O(q^{25})$

Coefficient field

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

$x + 74$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 52$ )
13	$-1$

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	17	−91	−233	134	169	797	−1310	−2016	−1290	8780	−7055	−17632	15247	−20581	−12928	26842	−30968	−54226	41317	34850	−20536	8556	−121510	144982

$\operatorname{tr} a_n$ 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 52, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.236 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, 4, 13, 26, 52]: [1, 2, 4, 7, 19, 38] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; 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(52, 6)[1]   # 52.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 52:6 1000