Atlas › Modular forms › Weight 4 › Level 147 › 147.4.a › 147.4.a.k

Newform orbit 147.4.a.k

Trace form

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

$2q + 2q^{2} + 6q^{3} - 10q^{4} - 20q^{5} + 6q^{6} - 18q^{8} + 18q^{9} - 48q^{10} - 20q^{11} - 30q^{12} - 104q^{13} - 60q^{15} + 18q^{16} - 116q^{17} + 18q^{18} - 192q^{19} + 44q^{20} + 76q^{22} + 28q^{23} - 54q^{24} + O(q^{25})$

Coefficient field

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

$x^{2} + 58x + 41$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
7	— ( $7^2 \mid 147$ )

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	6	−20	0	−20	−104	−116	−192	28	296	104	−248	−20	−720	96	268	616	−16	−144	988	−104	−944	−1016	388	−488

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.132 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, 7, 21, 49, 147]: [1, 2, 3, 10, 15, 58] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 5 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(147, 4)[10]   # 147.4.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 147:4 1000