Properties

Label 203.9
Modulus $203$
Conductor $203$
Order $42$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(203, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([14,15]))
 
pari: [g,chi] = znchar(Mod(9,203))
 

Basic properties

Modulus: \(203\)
Conductor: \(203\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 203.t

\(\chi_{203}(4,\cdot)\) \(\chi_{203}(9,\cdot)\) \(\chi_{203}(51,\cdot)\) \(\chi_{203}(67,\cdot)\) \(\chi_{203}(93,\cdot)\) \(\chi_{203}(100,\cdot)\) \(\chi_{203}(109,\cdot)\) \(\chi_{203}(121,\cdot)\) \(\chi_{203}(149,\cdot)\) \(\chi_{203}(151,\cdot)\) \(\chi_{203}(158,\cdot)\) \(\chi_{203}(179,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 42.42.496897759422042196258605771077406782550407598249513303021389442457964675897236469.1

Values on generators

\((59,176)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{5}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 203 }(9, a) \) \(1\)\(1\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 203 }(9,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 203 }(9,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 203 }(9,·),\chi_{ 203 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 203 }(9,·)) \;\) at \(\; a,b = \) e.g. 1,2