Atlas › Modular forms › Weight 4 › Level 128 › 128.4.a › 128.4.a.d

Newform orbit 128.4.a.d

$q$ -expansion

$q + 2q^{3} + 6q^{5} + 20q^{7} - 23q^{9} + 14q^{11} + 54q^{13} + 12q^{15} - 66q^{17} + 162q^{19} + 40q^{21} + 172q^{23} + O(q^{25})$

Coefficient field

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

$x - 20$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 128$ )

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	2	6	20	14	54	−66	162	172	−2	−128	158	202	−298	−408	−690	−322	−298	202	−700	−418	744	−678	−82	−1122

$\operatorname{tr} a_n$ 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 128, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.123 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, 8, 16, 32, 64, 128]: [1, 2, 3, 5, 8, 14, 26, 50] (certified at ell = 2147483489)
T = [(3, 1), (5, 3)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 3 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(128, 4)[3]   # 128.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 128:4 1000