Atlas › Modular forms › Weight 2 › Level 966 › 966.2.a › 966.2.a.m

Newform orbit 966.2.a.m

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{41})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 2T_{11} + 3T_{19} + 4T_{29} + 5T_{31} + 6T_{13} + 7T_{17} + 8T_{37} + 9T_{41} + 10T_{43} + 11T_{47} + 12T_{53}$ , whose minimal polynomial is

$x^{2} - 345x - 3546$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	$-1$
7	$+1$
23	$+1$

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
$\operatorname{tr} a_p$	−2	2	−1	−2	0	5	0	2	−2	7	12	1	9	5	1	6	−14	20	8	2	2	0	−18	−18	17

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.387 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, 7, 14, 21, 23, 42, 46, 69, 138, 161, 322, 483, 966]: [0, 1, 1, 3, 1, 4, 4, 3, 12, 8, 10, 28, 18, 52, 68, 200] (certified at ell = 2147483647)
T = [(5, 1), (11, 2), (19, 3), (29, 4), (31, 5), (13, 6), (17, 7), (37, 8), (41, 9), (43, 10), (47, 11), (53, 12)]; h of degree 21 = 1 x 21 (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(966, 2)[12]   # 966.2.a.m

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

cargo run --release -p sagebrush-web --example atlas -- 966:2 1000