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

Newform orbit 147.4.a.m

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + q^{2} + 9q^{3} + 25q^{4} - 11q^{5} + 3q^{6} + 39q^{8} + 27q^{9} + 55q^{10} + 35q^{11} + 75q^{12} - 62q^{13} - 33q^{15} + 241q^{16} - 48q^{17} + 9q^{18} + 202q^{19} - 439q^{20} - 7q^{22} + 216q^{23} + 117q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{3} + 32x^{2} - 1950x - 24852$

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
$\operatorname{tr} a_p$	1	9	−11	0	35	−62	−48	202	216	53	95	262	−244	360	210	393	−1143	70	−628	318	−988	861	−519	−1766	−19

$\operatorname{tr} a_n$ 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)[12]   # 147.4.a.m

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

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