Atlas › Modular forms › Weight 2 › Level 423 › 423.2.a › 423.2.a.j

Newform orbit 423.2.a.j

Trace form

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

$3q + 2q^{2} + 4q^{4} + q^{5} - 3q^{7} + 12q^{8} - 8q^{10} + 9q^{11} - 4q^{13} + 2q^{14} + 18q^{16} + 12q^{17} - 2q^{19} - 12q^{20} + 10q^{22} - q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{11}$ , whose minimal polynomial is

$x^{3} - 53x^{2} + 7x + 4157$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 423$ )
47	$-1$

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	1	−3	9	−4	12	−2	−1	9	−2	1	8	14	3	10	−2	−14	0	−18	14	−15	10	−2	−3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.313 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 3, 9, 47, 141, 423]: [0, 1, 2, 5, 18, 50] (certified at ell = 2147483137)
T = [(2, 1), (5, 3), (7, 5), (11, 7)]; h of degree 19 = 1 x 19 (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(423, 2)[9]   # 423.2.a.j

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

cargo run --release -p sagebrush-web --example atlas -- 423:2 1000