# char_dirichlet downloaded from the LMFDB on 13 June 2026.
# Search link: https://www.lmfdb.org/Character/Dirichlet/?modulus=41
# Query "{'modulus': 41}" returned 8 char_dirichlets, sorted by modulus.

# Each entry in the following data list has the form:
#    [Orbit label, Conrey labels, Modulus, Conductor, Order, Parity, Real, Primitive, Minimal]
# For more details, see the definitions at the bottom of the file.



"41.a"	[41, 1, 1, 1]	41	1	1	true	true	false	true
"41.b"	[41, 40, 40, 1]	41	41	2	true	true	true	true
"41.c"	[41, 9, 32, 2]	41	41	4	true	false	true	true
"41.d"	[41, 10, 37, 4]	41	41	5	true	false	true	true
"41.e"	[41, 3, 38, 4]	41	41	8	false	false	true	true
"41.f"	[41, 4, 31, 4]	41	41	10	true	false	true	true
"41.g"	[41, 2, 39, 8]	41	41	20	true	false	true	true
"41.h"	[41, 6, 35, 16]	41	41	40	false	false	true	true


#Orbit label (label) --
#    The **label** of a Galois orbit of a Dirichlet character $\chi$ of modulus $N$ takes the form $N.a$, where $a$ is a letter or string of letters representing the index of the Galois orbit.
#     The index $1$ is written as $a$, the index  $2$ is written as $b$, the index $27$ is written as $ba$, and so on.


#Conrey labels (conrey) --
#    We use the notation $\chi_{q}(n,\cdot)$ to identify Dirichlet characters $\Z\to \C$, where $q$ is the
#    modulus, and $n$ is the index, a positive integer coprime to $q$ that identifies a Dirichlet character of modulus $q$ as described below.  The LMFDB label $\texttt{q.n}$, with $1\le n < \max(q,2)$ uniquely identifies $\chi_{q}(n,\cdot)$.

#    Introduced by Brian Conrey, this labeling system is based on an explicit isomorphism between the multiplicative group $(\Z/q\Z)^\times$ and the group of Dirichlet characters of modulus $q$ that makes it easy to recover the
#    order, the
#    conductor, and the
#    parity
#    of a Dirichlet character from its label, or to induce characters.

#    As an example,
#    $\chi_q(1, \cdot)$ is always trivial, $\chi_q(m,\cdot)$ is real if $m^2=1\bmod q$, and for all $m,n$ coprime to $q$ we have $\chi_q(m,n)=\chi_q(n,m)$.

#    For prime powers $q=p^e$ we define $\chi_q(n,\cdot)$ as follows:

#    - For each odd prime $p$ we choose the least positive integer $g_p$ which
#     is a
#      primitive root for all $p^e$, and then for $n \equiv g_p^a $ mod $p^{e}$ and $m
#      \equiv g_p^{b} $ mod $p^{e}$ coprime to $p$ we define
#      $$
#        \chi_{p^e}(n, m) = \exp\left(2\pi i \frac{a b}{\phi(p^{e})} \right).
#      $$

#    - $\chi_2(1, \cdot)$ is trivial, $\chi_4(3, \cdot)$ is the
#      unique nontrivial character of modulus $4$, and for larger powers of $2$ we choose
#      $-1$ and $5$ as generators of the multiplicative group. For $e > 2$, if
#      $$
#        n \equiv \epsilon_a 5^a \pmod{2^e}
#      $$
#      and
#      $$
#        m \equiv \epsilon_b 5^b \pmod{2^e}
#      $$
#      with $\epsilon_a, \epsilon_b \in \{\pm 1\}$, then
#      \[
#        \chi_{2^e}(n, m) = \exp\left(2 \pi i \left(\frac{(1 - \epsilon_a)(1 - \epsilon_b)}{8}
#            + \frac{ab}{2^{e-2}}\right)\right).
#      \]

#    For general $q$, the function $\chi_q(n, m)$ is defined multiplicatively: $\chi_{q_1 q_2}(n, m) := \chi_{q_1}(n, m)\chi_{q_2}(n, m)$  for all coprime positive integers $q_1$ and $q_2$.  The Chinese remainder theorem implies that this definition is well founded and that every Dirichlet character can be defined in this way.  In particular, every Dirichlet character $\chi$ of modulus $q$ can be written uniquely as a product of Dirichlet characters of prime power modulus.




# Modulus --
#    A **Dirichlet character** is a function $\chi: \mathbb Z\to \mathbb C$ together with a positive integer $q$, called the **modulus** of the character, such that $\chi$ is completely multiplicative, i.e. $\chi(mn)=\chi(m)\chi(n)$ for all integers $m$ and $n$, and $\chi$ is periodic modulo $q$, i.e. $\chi(n+q)=\chi(n)$ for all $n$. If $(n,q)>1$ then $\chi(n)=0$, whereas if $(n,q)=1$, then $\chi(n)$ is a root of unity.


# Conductor --
#    The **conductor** of a Dirichlet character $\chi$ modulo $q$ is the least positive integer $q_1$ dividing $q$ for which $\chi(n+kq_1)=\chi(n)$ for all $n$ and $n+kq_1$ coprime to $q$.


# Order --
#    The **order** of a Dirichlet character $\chi$ is the least positive integer $n$ such that $\chi^n$ is the trivial character of the same modulus as $\chi$. Equivalently, it is the order $n$ of the image of $\chi$ in $\mathbb{C}^\times$, the group of $n$th roots of unity.


#Parity (is_even) --
#    A character has odd/even **parity** if it is odd/even as a function. A Dirichlet character $\chi\colon\Z\to\C$ is odd if $\chi(-1) = -1$ and even if $\chi(-1)=1.$


#Real (is_real) --
#    A character is **real** if its image is contained in $\mathbb R.$


#Primitive (is_primitive) --
#    A Dirichlet character $\chi$ is **primitive** if its
#    conductor is equal to its modulus; equivalently, $\chi$ is not induced by a Dirichlet character of smaller modulus.


#Minimal (is_minimal) --
#    A Dirichlet character $\chi$ of prime power modulus $N$ is **minimal** if the following conditions both hold:

#    1. The conductor of $\chi$ does not lie in the open interval $(\sqrt{N},N)$, and if $N$ is a square divisible by 16 then ${\rm cond}(\chi)\in \{\sqrt{N},N\}$.

#    2. Both the order and conductor of $\chi$ are minimal among the set of all Dirichlet character $\chi\psi^2$ for which ${\rm cond}(\psi){\rm cond}(\chi\psi) | N$.

#    This includes all primitive Dirichlet characters of prime power modulus, but not every minimal Dirichlet character of prime power modulus is primitive.

#    For a composite modulus $N$ with prime power factorization $N=p_1^{e_1}\cdots p_n^{e_n}$, a Dirichlet character $\chi$ of modulus $N$ is **minimal** if and only if every character in its unique factorization into Dirichlet characters of modulus $p_1^{e_1},\cdots,p_n^{e_n}$ is minimal.  The trivial Dirichlet character is minimal.