Properties

Label 203.199
Modulus $203$
Conductor $203$
Order $42$
Real no
Primitive yes
Minimal yes
Parity odd

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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([7,24]))
 
pari: [g,chi] = znchar(Mod(199,203))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 203.v

\(\chi_{203}(24,\cdot)\) \(\chi_{203}(45,\cdot)\) \(\chi_{203}(52,\cdot)\) \(\chi_{203}(54,\cdot)\) \(\chi_{203}(82,\cdot)\) \(\chi_{203}(94,\cdot)\) \(\chi_{203}(103,\cdot)\) \(\chi_{203}(110,\cdot)\) \(\chi_{203}(136,\cdot)\) \(\chi_{203}(152,\cdot)\) \(\chi_{203}(194,\cdot)\) \(\chi_{203}(199,\cdot)\)

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: 42.0.16778739246697564329550246936340186720321059686137149293129858772813957238199098503.1

Values on generators

\((59,176)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{4}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 203 }(199, a) \) \(-1\)\(1\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{5}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 203 }(199,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 203 }(199,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 203 }(199,·),\chi_{ 203 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 203 }(199,·)) \;\) at \(\; a,b = \) e.g. 1,2