Atlas › Modular forms › Weight 4 › Level 204 › 204.4.a › 204.4.a.a

Newform orbit 204.4.a.a

$q$ -expansion

$q + 3q^{3} - 3q^{5} - 16q^{7} + 9q^{9} - 57q^{11} - 25q^{13} - 9q^{15} + 17q^{17} - 13q^{19} - 48q^{21} - 93q^{23} + O(q^{25})$

Coefficient field

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

$x + 19$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 204$ )
3	$-1$
17	$-1$

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	3	−3	−16	−57	−25	17	−13	−93	−6	110	248	−333	−115	−294	−318	−30	668	−220	540	1214	−442	−438	60	1568

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 204, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.16 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, 12, 17, 34, 51, 68, 102, 204]: [1, 2, 2, 3, 5, 9, 6, 16, 20, 30, 58, 114] (certified at ell = 2147483489)
T = [(5, 1), (7, 1)]; h of degree 8 = 1 x 8 (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(204, 4)[0]   # 204.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 204:4 1000