Atlas › Modular forms › Weight 2 › Level 166 › 166.2.a › 166.2.a.c

Newform orbit 166.2.a.c

Trace form

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

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

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

$x^{3} - 43x^{2} - 781x + 5759$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
83	$+1$

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	1	−1	2	5	−9	−4	−5	7	13	−4	−19	11	3	2	−9	7	−15	15	6	6	12	−3	4	−10

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.177 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, 83, 166]: [0, 1, 8, 23] (certified at ell = 2147483323)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 6 = 1 x 6 (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(166, 2)[2]   # 166.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 166:2 1000