Editing Dirichlet character (section)

== The group of characters ==

There are three different cases because the groups <math>(\mathbb{Z}/m\mathbb{Z})^\times</math>  have different structures depending on whether <math>m</math> is a power of 2, a power of an odd prime, or the product of prime powers.<ref>Except for the use of the modified Conrie labeling, this section follows  Davenport pp. 1-3, 27-30</ref>

=== Powers of odd primes ===

If <math>q=p^k</math> is an odd number <math>(\mathbb{Z}/q\mathbb{Z})^\times</math> is cyclic of order <math>\phi(q)</math>; a generator is called a [[Primitive root modulo n|primitive root]] mod <math>q</math>.<ref>There is a primitive root mod <math>p</math> which is a primitive root mod <math>p^2</math> and all higher powers of <math>p</math>. See, e.g., Landau p. 106</ref>
Let <math>g_q</math> be a primitive root and  for <math>(a,q)=1</math> define the function <math>\nu_q(a)</math> (the '''index''' of <math>a</math>) by
:<math>a\equiv g_q^{\nu_q(a)}\pmod {q},</math>
:<math>0\le\nu_q<\phi(q).</math>
For <math>(ab,q)=1,\;\;a \equiv b\pmod{q}</math> if and only if <math>\nu_q(a)=\nu_q(b).</math> Since
:<math>\chi(a)=\chi(g_q^{\nu_q(a)})=\chi(g_q)^{\nu_q(a)},</math> &nbsp; <math>\chi</math> is determined by its value at <math>g_q.</math>

Let <math>\omega_q= \zeta_{\phi(q)}</math>  be a primitive <math>\phi(q)</math>-th root of unity. From property 7) above the possible values of <math> \chi(g_q)</math> are
<math> \omega_q, \omega_q^2, ... \omega_q^{\phi(q)}=1.</math> These distinct values give rise to  <math>\phi(q)</math> Dirichlet characters mod <math>q.</math> For <math>(r,q)=1</math> define <math>\chi_{q,r}(a)</math> as
:<math>
\chi_{q,r}(a)=
\begin{cases}
0 &\text{if } \gcd(a,q)>1\\
\omega_q^{\nu_q(r)\nu_q(a)}&\text{if } \gcd(a,q)=1.
\end{cases}</math>

Then for <math>(rs,q)=1</math> and all <math>a</math> and <math>b</math>
:<math>\chi_{q,r}(a)\chi_{q,r}(b)=\chi_{q,r}(ab),</math> showing that <math>\chi_{q,r}</math> is a character and
:<math>\chi_{q,r}(a)\chi_{q,s}(a)=\chi_{q,rs}(a),</math> which gives an explicit isomorphism <math>\widehat{(\mathbb{Z}/p^k\mathbb{Z})^\times}\cong(\mathbb{Z}/p^k\mathbb{Z})^\times.</math>

==== Examples ''m'' = 3, 5, 7, 9 ====
2 is a primitive root mod 3. &nbsp; (<math>\phi(3)=2</math>)
:<math>2^1\equiv 2,\;2^2\equiv2^0\equiv 1\pmod{3},</math>
so the values of <math>\nu_3</math> are
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
a & 1 & 2  \\
\hline
\nu_3(a) & 0 &  1\\
\end{array}
</math>.
The nonzero values of the characters mod 3 are
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
             & 1 & 2  \\
\hline
\chi_{3,1} & 1 & 1  \\
\chi_{3,2} & 1 & -1  \\
\end{array}
</math>

2 is a primitive root mod 5. &nbsp; (<math>\phi(5)=4</math>)
:<math>2^1\equiv 2,\;2^2\equiv 4,\;2^3\equiv 3,\;2^4\equiv2^0\equiv 1\pmod{5},</math>
so the values of <math>\nu_5</math> are
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
a & 1 & 2 & 3 & 4  \\
\hline
\nu_5(a) & 0 & 1 & 3 & 2 \\
\end{array}
</math>.
The nonzero values of the characters mod 5 are
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
             & 1 & 2  & 3    &  4  \\
\hline
\chi_{5,1} & 1 & 1   & 1   &  1 \\
\chi_{5,2} & 1 & i   & -i  & -1\\
\chi_{5,3} & 1 & -i  & i   & -1\\
\chi_{5,4} & 1 & -1  & -1  &  1\\
\end{array}
</math>

3 is a primitive root mod 7. &nbsp; (<math>\phi(7)=6</math>)
:<math>3^1\equiv 3,\;3^2\equiv 2,\;3^3\equiv 6,\;3^4\equiv 4,\;3^5\equiv 5,\;3^6\equiv3^0\equiv 1\pmod{7},</math>
so the values of <math>\nu_7</math> are
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
a        & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
\nu_7(a) & 0 & 2 & 1 & 4 & 5 & 3 \\
\end{array}
</math>.
The nonzero values of the characters mod 7 are (<math>\omega=\zeta_6, \;\;\omega^3=-1</math>)
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
             & 1 & 2         & 3         & 4        & 5          & 6   \\
\hline
\chi_{7,1} & 1 & 1         & 1         & 1        & 1          & 1   \\
\chi_{7,2} & 1 & -\omega   & \omega^2  & \omega^2 & -\omega    & 1   \\
\chi_{7,3} & 1 & \omega^2  & \omega    & -\omega  & -\omega^2  & -1  \\
\chi_{7,4} & 1 & \omega^2  & -\omega   & -\omega  & \omega^2   & 1   \\
\chi_{7,5} & 1 & -\omega   & -\omega^2 & \omega^2 & \omega     & -1  \\
\chi_{7,6} & 1 & 1         & -1        & 1        & -1         & -1  \\
\end{array}
</math>.

2 is a primitive root mod 9. &nbsp; (<math>\phi(9)=6</math>)
:<math>2^1\equiv 2,\;2^2\equiv 4,\;2^3\equiv 8,\;2^4\equiv 7,\;2^5\equiv 5,\;2^6\equiv2^0\equiv 1\pmod{9},</math>
so the values of <math>\nu_9</math> are
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
a & 1 & 2 &4 & 5&7&8 \\
\hline
\nu_9(a) & 0 & 1 & 2 & 5&4&3 \\
\end{array}
</math>.
The nonzero values of the characters mod 9 are (<math>\omega=\zeta_6, \;\;\omega^3=-1</math>)
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
               & 1 & 2         & 4         & 5         &7          & 8 \\
\hline
\chi_{9,1} & 1 & 1         & 1         & 1         & 1         & 1 \\
\chi_{9,2} & 1 & \omega    & \omega^2  & -\omega^2 & -\omega   & -1 \\
\chi_{9,4} & 1 & \omega^2  & -\omega   & -\omega   & \omega^2  & 1  \\
\chi_{9,5} & 1 & -\omega^2 & -\omega   & \omega    & \omega^2  & -1  \\
\chi_{9,7} & 1 & -\omega   &  \omega^2 & \omega^2  & -\omega   & 1  \\
\chi_{9,8} & 1 & -1        & 1         & -1        & 1         & -1  \\
\end{array}
</math>.

=== Powers of 2 ===

<math>(\mathbb{Z}/2\mathbb{Z})^\times</math> is the trivial group with one element. <math>(\mathbb{Z}/4\mathbb{Z})^\times</math> is cyclic of order 2. For 8, 16, and higher powers of 2, there is no primitive root; the powers of 5 are the units <math>\equiv 1\pmod{4}</math> and their negatives are the units <math>\equiv 3\pmod{4}.</math><ref>Landau pp. 107-108</ref>
For example
:<math>5^1\equiv 5,\;5^2\equiv5^0\equiv 1\pmod{8}</math>
:<math>5^1\equiv 5,\;5^2\equiv 9,\;5^3\equiv 13,\;5^4\equiv5^0\equiv 1\pmod{16}</math>
:<math>5^1\equiv 5,\;5^2\equiv 25,\;5^3\equiv 29,\;5^4\equiv 17,\;5^5\equiv 21,\;5^6\equiv 9,\;5^7\equiv 13,\;5^8\equiv5^0\equiv 1\pmod{32}.</math>
Let <math>q=2^k, \;\;k\ge3</math>; then <math>(\mathbb{Z}/q\mathbb{Z})^\times</math> is the direct product of a cyclic group of order 2 (generated by −1) and a cyclic group of order <math>\frac{\phi(q)}{2}</math> (generated by 5).
For odd numbers <math>a</math> define the functions <math>\nu_0</math> and <math>\nu_q</math> by
:<math>a\equiv(-1)^{\nu_0(a)}5^{\nu_q(a)}\pmod{q},</math>
:<math>0\le\nu_0<2,\;\;0\le\nu_q<\frac{\phi(q)}{2}.</math>
For odd <math>a</math> and <math>b, \;\;a\equiv b\pmod{q}</math> if and only if <math>\nu_0(a)=\nu_0(b)</math> and <math>\nu_q(a)=\nu_q(b).</math>
For odd <math>a</math> the value of <math> \chi(a)</math> is determined by the values of <math> \chi(-1)</math> and <math>\chi(5).</math>

Let <math>\omega_q = \zeta_{\frac{\phi(q)}{2}}</math> be a primitive <math>\frac{\phi(q)}{2}</math>-th root of unity. The possible values of <math> \chi((-1)^{\nu_0(a)}5^{\nu_q(a)})</math> are
<math> \pm\omega_q, \pm\omega_q^2, ... \pm\omega_q^{\frac{\phi(q)}{2}}=\pm1.</math> These distinct values give rise to  <math>\phi(q)</math> Dirichlet characters mod <math>q.</math> For odd <math>r </math> define <math>\chi_{q,r}(a)</math> by
:<math>
\chi_{q,r}(a)=
\begin{cases}
0 &\text{if } a\text{ is even}\\
(-1)^{\nu_0(r)\nu_0(a)}\omega_q^{\nu_q(r)\nu_q(a)}&\text{if } a \text{ is odd}.
\end{cases}</math>
Then for odd <math>r</math> and <math>s</math> and all <math>a</math> and <math>b</math>
:<math>\chi_{q,r}(a)\chi_{q,r}(b)=\chi_{q,r}(ab)</math> showing that <math>\chi_{q,r}</math> is a character and
:<math>\chi_{q,r}(a)\chi_{q,s}(a)=\chi_{q,rs}(a)</math> showing that <math>\widehat{(\mathbb{Z}/2^{k}\mathbb{Z})^\times}\cong (\mathbb{Z}/2^{k}\mathbb{Z})^\times.</math>

==== Examples ''m'' = 2, 4, 8, 16 ====

The only character mod 2 is the principal character <math>\chi_{2,1}</math>.

−1 is a primitive root mod 4 (<math>\phi(4)=2</math>)

:<math>
\begin{array}{|||}
    a    & 1 & 3  \\
\hline
\nu_0(a) & 0 & 1  \\
\end{array}
</math>

The  nonzero values of the characters mod 4 are
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
             & 1 & 3    \\
\hline
\chi_{4,1} & 1 & 1        \\
\chi_{4,3} & 1 & -1  \\

\end{array}
</math>

−1 is and 5 generate the units mod 8 (<math>\phi(8)=4</math>)

:<math>
\begin{array}{|||}
    a    & 1 & 3 & 5 & 7  \\
\hline
\nu_0(a) & 0 & 1 & 0 & 1 \\
\nu_8(a) & 0 & 1 & 1 & 0  \\
\end{array}
</math>.

The  nonzero values of the characters mod 8 are
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
             & 1 & 3  &  5  & 7     \\
\hline
\chi_{8,1} & 1 & 1  &  1  & 1      \\
\chi_{8,3} & 1 & 1  & -1  & -1 \\
\chi_{8,5} & 1 & -1 & -1  & 1    \\
\chi_{8,7} & 1 & -1 & 1   & -1     \\
\end{array}
</math>

−1 and 5 generate the units mod 16 (<math>\phi(16)=8</math>)
:<math>
\begin{array}{|||}
    a    & 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 \\
\hline
\nu_0(a) & 0 & 1 & 0 & 1 & 0 & 1  & 0  & 1  \\
\nu_{16}(a) & 0 & 3 & 1 & 2 & 2 & 1  & 3  & 0  \\
\end{array}
</math>.

The  nonzero values of the characters mod 16 are

:<math>
\begin{array}{|||}
                   & 1 & 3  & 5  & 7  & 9  & 11  & 13 & 15 \\
\hline
\chi_{16,1}    & 1 & 1  & 1  & 1  & 1  & 1   & 1  & 1  \\
\chi_{16,3}    & 1 & -i & -i & 1  & -1 & i   & i  & -1   \\
\chi_{16,5}    & 1 & -i & i  & -1 & -1 & i   & -i & 1  \\
\chi_{16,7}    & 1 & 1  & -1 & -1 & 1  & 1   & -1 & -1  \\
\chi_{16,9}    & 1 & -1 & -1 & 1  & 1  & -1  & -1 & 1  \\
\chi_{16,11}   & 1 & i  & i  & 1  & -1 & -i  & -i & -1  \\
\chi_{16,13}   & 1 & i  & -i & -1 & -1 & -i  & i  & 1  \\
\chi_{16,15}   & 1 & -1 & 1  & -1 & 1  & -1  & 1  & -1  \\

\end{array}
</math>.

=== Products of prime powers ===
Let <math>m=p_1^{m_1}p_2^{m_2} \cdots p_k^{m_k} = q_1q_2 \cdots q_k</math> where <math> p_1<p_2< \dots < p_k</math> be the factorization of <math>m</math> into prime powers. The group of units mod <math>m</math> is isomorphic to the direct product of the groups mod the <math>q_i</math>:<ref>See [[Multiplicative group of integers modulo n#General composite numbers|group of units]] for details</ref>
:<math>(\mathbb{Z}/m\mathbb{Z})^\times \cong(\mathbb{Z}/q_1\mathbb{Z})^\times \times(\mathbb{Z}/q_2\mathbb{Z})^\times \times \dots \times(\mathbb{Z}/q_k\mathbb{Z})^\times .</math>
This means that 1) there is a one-to-one correspondence between <math>a\in (\mathbb{Z}/m\mathbb{Z})^\times</math> and <math>k</math>-tuples <math>(a_1, a_2,\dots, a_k)</math> where <math>a_i\in(\mathbb{Z}/q_i\mathbb{Z})^\times </math> 
and 2) multiplication mod <math>m</math> corresponds to coordinate-wise multiplication of <math>k</math>-tuples:  
:<math>ab\equiv c\pmod{m}</math> corresponds to 
:<math>(a_1,a_2,\dots,a_k)\times(b_1,b_2,\dots,b_k)=(c_1,c_2,\dots,c_k)</math> where <math>c_i\equiv a_ib_i\pmod{q_i}.</math>

The [[Chinese remainder theorem]] (CRT) implies that the <math>a_i</math> are simply <math>a_i\equiv a\pmod{q_i}.</math>

There are subgroups <math> G_i<(\mathbb{Z}/m\mathbb{Z})^\times</math> such that <ref>To construct the <math>G_i, </math> for each <math> a\in (\mathbb{Z}/q_i\mathbb{Z})^\times </math> use the CRT to find <math>a_i\in (\mathbb{Z}/m\mathbb{Z})^\times</math> where
:<math>a_i\equiv 
\begin{cases}
a &\mod q_i\\
1&\mod q_j, j\ne i.
\end{cases}
</math>
</ref>
:<math>G_i\cong(\mathbb{Z}/q_i\mathbb{Z})^\times </math> and
:<math>G_i\equiv
\begin{cases}
(\mathbb{Z}/q_i\mathbb{Z})^\times &\mod q_i\\
\{1\}&\mod q_j, j\ne i.
\end{cases}
</math>

Then <math>(\mathbb{Z}/m\mathbb{Z})^\times \cong G_1\times G_2\times...\times G_k</math>
and every <math>a\in (\mathbb{Z}/m\mathbb{Z})^\times</math> corresponds to a <math>k</math>-tuple  <math>(a_1, a_2,...a_k)</math> where <math>a_i\in G_i </math> and <math>a_i\equiv a\pmod{q_i}. </math>  
Every <math>a\in (\mathbb{Z}/m\mathbb{Z})^\times</math> can be uniquely factored as <math>a =a_1a_2...a_k.</math>
<ref>Assume <math>a</math> corresponds to <math>(a_1,a_2, ...)</math>. By construction <math>a_1</math> corresponds to <math>(a_1,1,1,...)</math>, <math>a_2</math> to <math>(1,a_2,1,...)</math> etc. whose coordinate-wise product is <math>(a_1,a_2, ...).</math></ref>
<ref>For example let <math>m=40, q_1=8, q_2=5.</math> Then <math>G_1=\{1,11,21,31\}</math> and <math>G_2=\{1,9,17,33\}.</math> The factorization of the elements of <math>(\mathbb{Z}/40\mathbb{Z})^\times</math> is 

:<math>
\begin{array}{|c|c|c|c|c|c|c|}
     & 1  &  9   &  17  & 33  \\
\hline
  1  & 1  &  9   &  17  & 33  \\
  11 & 11 &  19  &  27  & 3   \\
  21 & 21 &  29  &  37  & 13  \\
  31 & 31 &  39  &   7  & 23  \\
\end{array}
</math>
</ref>

If <math>\chi_{m,\_}</math> is a character mod <math>m,</math> on the subgroup <math>G_i</math> it must be identical to some <math>\chi_{q_i,\_}</math> mod <math>q_i</math> Then
:<math>\chi_{m,\_}(a)=\chi_{m,\_}(a_1a_2...)=\chi_{m,\_}(a_1)\chi_{m,\_}(a_2)...=\chi_{q_1,\_}(a_1)\chi_{q_2,\_}(a_2)...,</math>
showing that every character mod <math> m</math> is the product of characters mod the <math>q_i</math>. 

For <math>(t,m)=1</math> define<ref>See [https://lmfdb.org/knowledge/show/character.dirichlet.conrey Conrey labeling].</ref>
:<math> \chi_{m,t}=\chi_{q_1,t}\chi_{q_2,t}...</math>
Then for <math>(rs,m)=1</math> and all <math>a</math> and <math>b</math><ref>Because these formulas are true for each factor.</ref>
:<math>\chi_{m,r}(a)\chi_{m,r}(b)=\chi_{m,r}(ab),</math> showing that <math>\chi_{m,r}</math> is a character and
:<math>\chi_{m,r}(a)\chi_{m,s}(a)=\chi_{m,rs}(a),</math> showing an isomorphism <math>\widehat{(\mathbb{Z}/m\mathbb{Z})^\times}\cong(\mathbb{Z}/m\mathbb{Z})^\times.</math>


==== Examples ''m'' = 15, 24, 40 ====

<math>(\mathbb{Z}/15\mathbb{Z})^\times\cong(\mathbb{Z}/3\mathbb{Z})^\times\times(\mathbb{Z}/5\mathbb{Z})^\times.</math>

The factorization of the characters mod 15 is
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
             & \chi_{5,1} &  \chi_{5,2} &  \chi_{5,3} &  \chi_{5,4}   \\
\hline
\chi_{3,1} & \chi_{15,1} & \chi_{15,7} & \chi_{15,13} & \chi_{15,4}  \\
\chi_{3,2} & \chi_{15,11} & \chi_{15,2} & \chi_{15,8} & \chi_{15,14} \\
\end{array} </math>
The nonzero values of the characters mod 15 are

:<math>
\begin{array}{|||}
                 & 1 & 2   & 4   & 7   & 8    & 11  & 13 & 14 \\
\hline
\chi_{15,1}    & 1 & 1   & 1   & 1   & 1    & 1   & 1   & 1  \\
\chi_{15,2}    & 1 & -i  & -1  & i   & i   & -1  & -i  & 1   \\
\chi_{15,4}    & 1 & -1  & 1   & -1  & -1   & 1   & -1  & 1  \\
\chi_{15,7}    & 1 &  i  & -1  & i   & -i   & 1   & -i  & -1  \\
\chi_{15,8}    & 1 &  i  & -1  & -i  & -i   & -1  & i   & 1  \\
\chi_{15,11}   & 1 & -1  & 1   & 1   &  -1  & -1  & 1  & -1  \\
\chi_{15,13}   & 1 & -i   & -1  & -i  & i   & 1  & i   & -1  \\
\chi_{15,14}   & 1 & 1   & 1   & -1   & 1    & -1  & -1   & -1  \\

\end{array}
</math>.

<math>(\mathbb{Z}/24\mathbb{Z})^\times\cong(\mathbb{Z}/8\mathbb{Z})^\times\times(\mathbb{Z}/3\mathbb{Z})^\times.</math>
The factorization of the characters mod 24 is
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
             & \chi_{8,1} &  \chi_{8,3} &  \chi_{8,5} &  \chi_{8,7}   \\
\hline
\chi_{3,1} & \chi_{24,1} & \chi_{24,19} & \chi_{24,13} & \chi_{24,7}  \\
\chi_{3,2} & \chi_{24,17} & \chi_{24,11} & \chi_{24,5} & \chi_{24,23} \\
\end{array} </math>

The nonzero values of the characters mod 24 are

:<math>
\begin{array}{|||}
                 & 1 & 5   & 7   & 11   & 13    & 17  & 19 & 23 \\
\hline
\chi_{24,1}    & 1 & 1   & 1   & 1   & 1    & 1   & 1   & 1  \\
\chi_{24,5}    & 1 & 1  & 1  & 1   & -1   & -1  & -1  & -1   \\
\chi_{24,7}    & 1 & 1  & -1   & -1  & 1   & 1   & -1  & -1  \\
\chi_{24,11}    & 1 &  1  & -1  & -1   & -1   & -1   & 1  & 1  \\
\chi_{24,13}    & 1 &  -1  & 1  & -1  & -1   & 1  & -1   & 1  \\
\chi_{24,17}   & 1 & -1  & 1   & -1   &  1  & -1  & 1  & -1  \\
\chi_{24,19}   & 1 & -1   & -1  & 1  & -1   & 1  & 1   & -1  \\
\chi_{24,23}   & 1 & -1   & -1   & 1   & 1    & -1  & -1   & 1  \\

\end{array}
</math>.

<math>(\mathbb{Z}/40\mathbb{Z})^\times\cong(\mathbb{Z}/8\mathbb{Z})^\times\times(\mathbb{Z}/5\mathbb{Z})^\times.</math>
The factorization of the characters mod 40 is
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
             & \chi_{8,1}   &  \chi_{8,3}  &  \chi_{8,5}  &  \chi_{8,7}   \\
\hline
\chi_{5,1} & \chi_{40,1}  & \chi_{40,11} & \chi_{40,21} & \chi_{40,31}  \\
\chi_{5,2} & \chi_{40,17} & \chi_{40,27} & \chi_{40,37} & \chi_{40,7} \\
\chi_{5,3} & \chi_{40,33} & \chi_{40,3}  & \chi_{40,13} & \chi_{40,23}  \\
\chi_{5,4} & \chi_{40,9}  & \chi_{40,19} & \chi_{40,29} & \chi_{40,39} \\
\end{array} </math>

The nonzero values of the characters mod 40 are

:<math>
\begin{array}{|||}
                 & 1 & 3  & 7   & 9   & 11   & 13  & 17 & 19 & 21 & 23  & 27  & 29 & 31 & 33  & 37 & 39 \\
\hline
\chi_{40,1}    & 1 & 1  & 1   & 1   & 1    & 1   & 1  & 1  & 1  & 1   & 1   & 1  & 1  & 1   & 1  & 1  \\
\chi_{40,3}    & 1 & i  & i   & -1  & 1    & -i  & -i & -1 & -1 & -i  & -i  & 1  & -1 & i   & i  & 1  \\
\chi_{40,7}    & 1 & i  & -i  & -1  & -1   & -i  & i  & 1  & 1  & i   & -i  & -1 & -1 & -i  & i  & 1  \\
\chi_{40,9}    & 1 & -1 & -1  & 1   & 1    & -1  & -1 & 1  & 1  & -1  & -1  &  1 & 1  & -1  & -1 & 1  \\
\chi_{40,11}   & 1 & 1  & -1  & 1   & 1    & -1  & 1  & 1  & -1 & -1  & 1   & -1 & -1 & 1   & -1 & -1 \\
\chi_{40,13}   & 1 & -i & -i  & -1  & -1   & -i  & -i & 1  & -1 & i   & i   & 1  &  1 & i   & i  & -1 \\
\chi_{40,17}   & 1 & -i & i   & -1  & 1    & -i  & i  & -1 & 1  & -i  & i   & -1 &  1 & -i  & i  & -1 \\
\chi_{40,19}   & 1 & -1 & 1   & 1   & 1    & 1   & -1 & 1  & -1 & 1   & -1  & -1 & -1 & -1  & 1 & -1  \\
\chi_{40,21}   & 1 & -1 & 1   & 1   & -1   & -1  & 1  & -1 & -1 & 1   & -1  & -1 &  1 & 1   & -1 & 1  \\
\chi_{40,23}   & 1 & -i & i   & -1  & -1   & i   & -i & 1  & 1  & -i  & i   & -1 & -1 & i   & -i & 1  \\
\chi_{40,27}   & 1 & -i & -i  & -1  & 1    & i   & i  & -1 & -1 & i   & i   & 1  & -1 & -i  & -i & 1  \\
\chi_{40,29}   & 1 & 1  & -1  & 1   & -1   & 1   & -1 & -1 & -1 & -1  & 1   & -1 &  1 & -1  & 1  & 1  \\
\chi_{40,31}   & 1 & -1 & -1  & 1   & -1   & 1   & 1  & -1 & 1  & -1  & -1  & 1  & -1 & 1   & 1  & -1 \\
\chi_{40,33}   & 1 & i  & -i  & -1  & 1    & i   & -i & -1 &  1 & i   & -i  & -1 & 1  & i   & -i & -1 \\
\chi_{40,37}   & 1 & i  & i   & -1  & -1   & i   & i  & 1  & -1 & -i  & -i  &  1 &  1 & -i  & -i & -1 \\
\chi_{40,39}   & 1 & 1  & 1   & 1   & -1   & -1  & -1 & -1 & 1  & 1   & 1   & 1  & -1 & -1  & -1 & -1 \\
\end{array}
</math>.

=== Summary ===
Let <math>m=p_1^{k_1}p_2^{k_2}\cdots = q_1q_2 \cdots</math>, <math>p_1<p_2< \dots </math> be the factorization of <math>m</math> and assume <math>(rs,m)=1.</math>

There are <math>\phi(m)</math> Dirichlet characters mod <math>m.</math> They are denoted by <math>\chi_{m,r},</math> where <math>\chi_{m,r}=\chi_{m,s}</math> is equivalent to <math>r\equiv s\pmod{m}.</math>
The identity <math>\chi_{m,r}(a)\chi_{m,s}(a)=\chi_{m,rs}(a)\;</math> is an isomorphism <math>\widehat{(\mathbb{Z}/m\mathbb{Z})^\times}\cong(\mathbb{Z}/m\mathbb{Z})^\times.</math><ref>This is true for all finite abelian groups: <math>A\cong\hat{A}</math>; See Ireland & Rosen pp. 253-254</ref>

Each character mod <math>m</math> has a unique factorization as the product of characters mod the prime powers dividing <math>m</math>:
:<math>\chi_{m,r}=\chi_{q_1,r}\chi_{q_2,r}...</math>

If <math>m=m_1m_2, (m_1,m_2)=1</math> the product <math>\chi_{m_1,r}\chi_{m_2,s}</math> is a character <math>\chi_{m,t}</math> where <math>t</math> is given by <math>t\equiv r\pmod{m_1}</math> and <math>t\equiv s\pmod{m_2}.</math>

Also,<ref>because the formulas for <math>\chi</math> mod prime powers are symmetric in <math>r</math> and <math>s</math> and the formula for products preserves this symmetry. See Davenport, p. 29.</ref><ref>This is the same thing as saying that the n-th column and the n-th row in the tables of nonzero values are the same.</ref>
<math> \chi_{m,r}(s)=\chi_{m,s}(r)</math>