Atlas › Modular forms › Weight 2 › Level 774 › 774.2.a › 774.2.a.l

Newform orbit 774.2.a.l

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{21})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{47}$ , whose minimal polynomial is

$x^{2} - 205x + 3319$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	— ( $3^2 \mid 774$ )
43	$-1$

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	−3	4	0	4	9	1	9	−3	1	1	−3	2	9	−6	−12	4	−20	0	28	−5	6	−6	−11

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.274 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, 6, 9, 18, 43, 86, 129, 258, 387, 774]: [0, 1, 1, 3, 2, 5, 4, 13, 16, 48, 46, 136] (certified at ell = 2147483647)
T = [(5, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (47, 12)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 6 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(774, 2)[11]   # 774.2.a.l

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

cargo run --release -p sagebrush-web --example atlas -- 774:2 1000