Atlas › Modular forms › Weight 4 › Level 99 › 99.4.a › 99.4.a.c

Newform orbit 99.4.a.c

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 2q^{2} - 8q^{4} - 2q^{5} + 20q^{7} + 12q^{8} + 50q^{10} + 22q^{11} + 80q^{13} + 4q^{14} - 8q^{16} + 124q^{17} + 72q^{19} - 88q^{20} - 22q^{22} + 98q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{2} + 8x - 1859$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 99$ )
11	$-1$

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
$\operatorname{tr} a_p$	−2	0	−2	20	22	80	124	72	98	−144	−34	54	−536	−60	272	492	−634	840	754	678	−400	316	−468	1842	2194

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.113 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, 3, 9, 11, 33, 99]: [1, 2, 4, 4, 14, 38] (certified at ell = 2147482951)
T = [(2, 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(99, 4)[2]   # 99.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 99:4 1000