Atlas › Modular forms › Weight 2 › Level 744 › 744.2.a › 744.2.a.i

Newform orbit 744.2.a.i

Trace form

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

$2q + 2q^{3} + 6q^{5} - q^{7} + 2q^{9} - 3q^{11} + 7q^{13} + 6q^{15} + 3q^{17} - q^{21} + 2q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{41})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17} + 11T_{19}$ , whose minimal polynomial is

$x^{2} - 64x - 8201$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 744$ )
3	$-1$
31	$+1$

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
$\operatorname{tr} a_p$	0	2	6	−1	−3	7	3	0	2	0	−2	−6	−1	−10	3	2	−1	−19	5	4	2	−9	5	20	−10

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.718 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, 2, 3, 4, 6, 8, 12, 24, 31, 62, 93, 124, 186, 248, 372, 744]: [0, 1, 1, 2, 3, 3, 5, 8, 3, 10, 12, 19, 36, 36, 70, 136] (certified at ell = 2147482951)
T = [(5, 1), (7, 3), (11, 5), (13, 7), (17, 9), (19, 11)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 4 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(744, 2)[8]   # 744.2.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 744:2 1000