Properties

Label 197.44
Modulus $197$
Conductor $197$
Order $196$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(197, base_ring=CyclotomicField(196))
 
M = H._module
 
chi = DirichletCharacter(H, M([31]))
 
pari: [g,chi] = znchar(Mod(44,197))
 

Basic properties

Modulus: \(197\)
Conductor: \(197\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(196\)
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 197.i

\(\chi_{197}(2,\cdot)\) \(\chi_{197}(3,\cdot)\) \(\chi_{197}(5,\cdot)\) \(\chi_{197}(8,\cdot)\) \(\chi_{197}(11,\cdot)\) \(\chi_{197}(12,\cdot)\) \(\chi_{197}(13,\cdot)\) \(\chi_{197}(17,\cdot)\) \(\chi_{197}(18,\cdot)\) \(\chi_{197}(21,\cdot)\) \(\chi_{197}(27,\cdot)\) \(\chi_{197}(30,\cdot)\) \(\chi_{197}(31,\cdot)\) \(\chi_{197}(32,\cdot)\) \(\chi_{197}(35,\cdot)\) \(\chi_{197}(38,\cdot)\) \(\chi_{197}(44,\cdot)\) \(\chi_{197}(45,\cdot)\) \(\chi_{197}(46,\cdot)\) \(\chi_{197}(48,\cdot)\) \(\chi_{197}(50,\cdot)\) \(\chi_{197}(52,\cdot)\) \(\chi_{197}(56,\cdot)\) \(\chi_{197}(57,\cdot)\) \(\chi_{197}(58,\cdot)\) \(\chi_{197}(66,\cdot)\) \(\chi_{197}(67,\cdot)\) \(\chi_{197}(71,\cdot)\) \(\chi_{197}(72,\cdot)\) \(\chi_{197}(73,\cdot)\) ...

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: $\Q(\zeta_{196})$
Fixed field: Number field defined by a degree 196 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{31}{196}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 197 }(44, a) \) \(-1\)\(1\)\(e\left(\frac{31}{196}\right)\)\(e\left(\frac{123}{196}\right)\)\(e\left(\frac{31}{98}\right)\)\(e\left(\frac{15}{196}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{9}{98}\right)\)\(e\left(\frac{93}{196}\right)\)\(e\left(\frac{25}{98}\right)\)\(e\left(\frac{23}{98}\right)\)\(e\left(\frac{115}{196}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 197 }(44,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 197 }(44,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 197 }(44,·),\chi_{ 197 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 197 }(44,·)) \;\) at \(\; a,b = \) e.g. 1,2