Atlas › Modular forms › Weight 4 › Level 98 › 98.4.a › 98.4.a.d

Newform orbit 98.4.a.d

$q$ -expansion

$q + 2q^{2} - 5q^{3} + 4q^{4} - 9q^{5} - 10q^{6} + 8q^{8} - 2q^{9} - 18q^{10} - 57q^{11} - 20q^{12} - 70q^{13} + 45q^{15} + 16q^{16} + 51q^{17} - 4q^{18} + 5q^{19} - 36q^{20} - 114q^{22} + 69q^{23} - 40q^{24} + O(q^{25})$

Coefficient field

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

$x + 32$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
7	— ( $7^2 \mid 98$ )

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
$\operatorname{tr} a_p$	2	−5	−9	0	−57	−70	51	5	69	114	23	−253	−42	−124	201	−393	219	−709	419	−96	−313	461	−588	−1017	−1834

$\operatorname{tr} a_n$ 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 98, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.117 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, 7, 14, 49, 98]: [1, 2, 3, 8, 15, 44] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 10 = 1 x 10 (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(98, 4)[3]   # 98.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 98:4 1000