Atlas › Modular forms › Weight 2 › Level 646 › 646.2.a › 646.2.a.k

Newform orbit 646.2.a.k

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 3q^{2} + 6q^{3} + 3q^{4} + 2q^{5} + 6q^{6} + 2q^{7} + 3q^{8} + 3q^{9} + 2q^{10} + 2q^{11} + 6q^{12} - 6q^{13} + 2q^{14} + 4q^{15} + 3q^{16} + 3q^{17} + 3q^{18} + 3q^{19} + 2q^{20} + 4q^{21} + 2q^{22} - 6q^{23} + 6q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} - 36x^{2} + 320x - 512$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
17	$-1$
19	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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
$\operatorname{tr} a_p$	3	6	2	2	2	−6	3	3	−6	2	−2	−6	2	−8	−8	−2	16	−18	4	10	−2	6	16	14	14

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.289 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, 17, 19, 34, 38, 323, 646]: [0, 1, 2, 2, 6, 7, 32, 94] (certified at ell = 2147482801)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 23 = 1 x 23 (phi(ord eps) x dim S_k^new); 5 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(646, 2)[10]   # 646.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 646:2 1000