# Properties

 Label 163.3.b.a Level $163$ Weight $3$ Character orbit 163.b Self dual yes Analytic conductor $4.441$ Analytic rank $0$ Dimension $1$ CM discriminant -163 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$163$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 163.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.44142830907$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4 q^{4} + 9 q^{9}+O(q^{10})$$ q + 4 * q^4 + 9 * q^9 $$q + 4 q^{4} + 9 q^{9} + 16 q^{16} + 25 q^{25} + 36 q^{36} - 81 q^{41} - 77 q^{43} - 69 q^{47} + 49 q^{49} - 57 q^{53} - 41 q^{61} + 64 q^{64} - 21 q^{71} + 81 q^{81} + 3 q^{83} + 31 q^{97}+O(q^{100})$$ q + 4 * q^4 + 9 * q^9 + 16 * q^16 + 25 * q^25 + 36 * q^36 - 81 * q^41 - 77 * q^43 - 69 * q^47 + 49 * q^49 - 57 * q^53 - 41 * q^61 + 64 * q^64 - 21 * q^71 + 81 * q^81 + 3 * q^83 + 31 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/163\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
162.1
 0
0 0 4.00000 0 0 0 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
163.b odd 2 1 CM by $$\Q(\sqrt{-163})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 163.3.b.a 1
163.b odd 2 1 CM 163.3.b.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
163.3.b.a 1 1.a even 1 1 trivial
163.3.b.a 1 163.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{3}^{\mathrm{new}}(163, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T + 81$$
$43$ $$T + 77$$
$47$ $$T + 69$$
$53$ $$T + 57$$
$59$ $$T$$
$61$ $$T + 41$$
$67$ $$T$$
$71$ $$T + 21$$
$73$ $$T$$
$79$ $$T$$
$83$ $$T - 3$$
$89$ $$T$$
$97$ $$T - 31$$
The Fourier coefficient $a_{n^2}$ for this newform is $n^2$ for all $n < 41$, and $a_p = 0$ for $p < 41$, yielding a striking beginning to the $q$-expansion.