Atlas › Modular forms › Weight 2 › Level 684 › 684.2.a › 684.2.a.d

Newform orbit 684.2.a.d

Trace form

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

$2q - 3q^{5} + q^{7} + 3q^{11} + 4q^{13} + 3q^{17} + 2q^{19} + 6q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{33})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 2T_{7} + 3T_{13} + 4T_{17} + 5T_{23} + 6T_{11} + 7T_{29} + 8T_{31}$ , whose minimal polynomial is

$x^{2} - 97x - 3662$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 684$ )
3	— ( $3^2 \mid 684$ )
19	$-1$

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	0	−3	1	3	4	3	2	6	6	−2	−2	0	1	21	−6	0	−11	4	24	−5	16	−6	−18	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 0.507 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, 9, 12, 18, 19, 36, 38, 57, 76, 114, 171, 228, 342, 684]: [0, 1, 1, 2, 3, 2, 5, 5, 2, 9, 7, 8, 13, 24, 22, 46, 64, 126] (certified at ell = 2147483647)
T = [(5, 1), (7, 2), (13, 3), (17, 4), (23, 5), (11, 6), (29, 7), (31, 8)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 2 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(684, 2)[3]   # 684.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 684:2 1000