CoCalc -- Dirichlet Characters

# Dirichlet Characters
#
# Dirichlet characters are extensions to Z of multiplicative characters chi:(Z/nZ)*->R*
# The target ring R is usually a cyclotomic field Q(z), with z a root of unity, 
#     so R* is the corresponding group of roots of unity.
#
# Note: we avoid the continuous circle S^1, to keep everything finite! 
#       so think chi is valued in a finite circle
# 
G=DirichletGroup(35) # ring R=Z/35Z
G.gen(0).base_ring() # R is a cyclotomic field; see SAGE number fields

Cyclotomic Field of order 12 and degree 4

# ... this is Q(z12), and [Q(z12):Q]=4  (never mind this ... for now)
# Note: Z35*=Z5*xZ7* ...

# Generators of the group of Dirichlet characters
x=G.gens(); # generators
x[0]
x[1]

Dirichlet character modulo 35 of conductor 5 mapping 22 |--> zeta12^3, 31 |--> 1
Dirichlet character modulo 35 of conductor 7 mapping 22 |--> 1, 31 |--> zeta12^2

# Q: Why not z24? i.e. (5-1)x(7-1)=24

# Q: What is "22" anyways? A: e(1)=22 is an order 4=5-1 element in Z/35Z=Z5 x Z7, 
#    Z/35Z*=Z5*x Z7* iso to the additive group Z4 x Z6
[mod(22^k,35) for k in range(10)]

[1, 22, 29, 8, 1, 22, 29, 8, 1, 22]

# Similarly e(2)=31 is an element of order 6=7-1
[mod(31^k,35) for k in range(7)]

[1, 31, 16, 6, 11, 26, 1]

# Chinese Remainder Th. yields the above iso f(k)=(k mod 5, k mod 7): Z35 ->Z5xZ7
# The preimage of the standard basis (2,1), (1,2) in Z5*xZ7* gives e(1)=22, e(2)=31 mod 35
print mod(22,5), mod(22,7) # (2,1) and 2 is a generator of Z5*
[mod(2^k,5) for k in range(5)]
print mod(31,5), mod(31,7)
[mod(3^k,7) for k in range(7)]

2 1
[1, 2, 4, 3, 1]
1 3
[1, 3, 2, 6, 4, 5, 1]

# Note: 2 is not a generator in Z7*
[mod(2^k,7) for k in range(7)]

[1, 2, 4, 1, 2, 4, 1]

# ***************************************************************************************************************
# Prime case
G=DirichletGroup(7); G # ring Z/7Z => Z/7Z* is the additive group Z/6Z; so R=Z/7Z

Group of Dirichlet characters of modulus 7 over Cyclotomic Field of order 6 and degree 2

G.gen(0).base_ring()
list(G); len(list(G)) # the list of charcaters and its length
x=G.gens(); len(x) # the generators of the group of characters
x[0]
x[0].values() # Lists the values of a character

Cyclotomic Field of order 6 and degree 2
[Dirichlet character modulo 7 of conductor 1 mapping 3 |--> 1, Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6, Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6 - 1, Dirichlet character modulo 7 of conductor 7 mapping 3 |--> -1, Dirichlet character modulo 7 of conductor 7 mapping 3 |--> -zeta6, Dirichlet character modulo 7 of conductor 7 mapping 3 |--> -zeta6 + 1]
6
1
Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6
[0, 1, zeta6 - 1, zeta6, -zeta6, -zeta6 + 1, -1]

# The values of the character chi(k), k=0,...,n-1
e = DirichletGroup(20)(1) # modulus n=20, first character
e.values()

[0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1]

# Number of characters = size of Z/nZ)*
len(list(G)) # returns the size of a list (in general)

6

# analize the example ... add facts about G etc.