Atlas › Modular forms › Weight 2 › Level 163 › 163.2.a › 163.2.a.b

Newform orbit 163.2.a.b

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} + 10x^{4} + 32x^{3} + 41x^{2} + 20x + 3$

Atkin–Lehner signs

$p$	$w_p$
163	$+1$

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 odd 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
$\operatorname{tr} a_p$	−5	−5	−9	−6	2	−14	−21	7	−8	−13	7	−1	−9	−4	−3	−16	−3	−10	−2	31	−14	−14	−12	0	23

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.242 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, 163]: [0, 14] (certified at ell = 2147483179)
T = [(2, 1), (3, 1)]; h of degree 13 = 1 x 13 (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(163, 2)[1]   # 163.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 163:2 1000