Atlas › Modular forms › Weight 2 › Level 747 › 747.2.a › 747.2.a.k

Newform orbit 747.2.a.k

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 4q^{2} + 10q^{4} + 8q^{5} + 18q^{8} + 6q^{10} + 6q^{11} - 10q^{13} + 6q^{14} + 10q^{16} + 4q^{17} - 4q^{19} + 22q^{20} - 10q^{22} + 10q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{6} - 24x^{5} - 8437x^{4} + 159304x^{3} + 16897123x^{2} - 93594464x - 8165940343$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 747$ )
83	$-1$

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
$\operatorname{tr} a_p$	4	0	8	0	6	−10	4	−4	10	8	0	−6	8	−2	12	50	10	−2	22	22	−6	−18	6	−10	−6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.428 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, 83, 249, 747]: [0, 1, 2, 8, 30, 86] (certified at ell = 2147483323)
T = [(2, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 34 = 1 x 34 (phi(ord eps) x dim S_k^new); 9 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(747, 2)[10]   # 747.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 747:2 1000