Properties

Label 203.114
Modulus $203$
Conductor $203$
Order $84$
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(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([28,45]))
 
pari: [g,chi] = znchar(Mod(114,203))
 

Basic properties

Modulus: \(203\)
Conductor: \(203\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
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.w

\(\chi_{203}(2,\cdot)\) \(\chi_{203}(11,\cdot)\) \(\chi_{203}(18,\cdot)\) \(\chi_{203}(32,\cdot)\) \(\chi_{203}(37,\cdot)\) \(\chi_{203}(39,\cdot)\) \(\chi_{203}(44,\cdot)\) \(\chi_{203}(60,\cdot)\) \(\chi_{203}(72,\cdot)\) \(\chi_{203}(79,\cdot)\) \(\chi_{203}(95,\cdot)\) \(\chi_{203}(102,\cdot)\) \(\chi_{203}(114,\cdot)\) \(\chi_{203}(130,\cdot)\) \(\chi_{203}(135,\cdot)\) \(\chi_{203}(137,\cdot)\) \(\chi_{203}(142,\cdot)\) \(\chi_{203}(156,\cdot)\) \(\chi_{203}(163,\cdot)\) \(\chi_{203}(172,\cdot)\) \(\chi_{203}(177,\cdot)\) \(\chi_{203}(184,\cdot)\) \(\chi_{203}(193,\cdot)\) \(\chi_{203}(200,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((59,176)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{15}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 203 }(114, a) \) \(-1\)\(1\)\(e\left(\frac{17}{84}\right)\)\(e\left(\frac{1}{84}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{55}{84}\right)\)\(e\left(\frac{61}{84}\right)\)\(e\left(\frac{5}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 203 }(114,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 203 }(114,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 203 }(114,·),\chi_{ 203 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 203 }(114,·)) \;\) at \(\; a,b = \) e.g. 1,2