# 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 disc. -163 Inner twists 2

# Related objects

## Newspace parameters

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

## 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: $$1$$ Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

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

## 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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
163.b Odd 1 CM by $$\Q(\sqrt{-163})$$ yes

## Hecke kernels

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