Atlas › Modular forms › Weight 2 › Level 722 › 722.2.a › 722.2.a.n

Newform orbit 722.2.a.n

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 4q^{2} + 2q^{3} + 4q^{4} - 2q^{5} + 2q^{6} - 2q^{7} + 4q^{8} + 4q^{9} - 2q^{10} + 2q^{11} + 2q^{12} + 18q^{13} - 2q^{14} + 4q^{15} + 4q^{16} + 6q^{17} + 4q^{18} - 2q^{20} + 4q^{21} + 2q^{22} - 10q^{23} + 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ , whose minimal polynomial is

$x^{4} - 1525x^{2} + 3720x + 457805$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
19	— ( $19^2 \mid 722$ )

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	2	−2	−2	2	18	6	0	−10	−2	26	4	−12	−10	−12	8	−8	0	10	0	−14	22	−12	−16	28

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.403 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, 19, 38, 361, 722]: [0, 1, 2, 7, 32, 97] (certified at ell = 2147483323)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 7 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(722, 2)[13]   # 722.2.a.n

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

cargo run --release -p sagebrush-web --example atlas -- 722:2 1000