Properties

Label 197.112
Modulus $197$
Conductor $197$
Order $98$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(197, base_ring=CyclotomicField(98))
 
M = H._module
 
chi = DirichletCharacter(H, M([75]))
 
pari: [g,chi] = znchar(Mod(112,197))
 

Basic properties

Modulus: \(197\)
Conductor: \(197\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(98\)
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 197.h

\(\chi_{197}(4,\cdot)\) \(\chi_{197}(7,\cdot)\) \(\chi_{197}(9,\cdot)\) \(\chi_{197}(10,\cdot)\) \(\chi_{197}(15,\cdot)\) \(\chi_{197}(22,\cdot)\) \(\chi_{197}(25,\cdot)\) \(\chi_{197}(26,\cdot)\) \(\chi_{197}(39,\cdot)\) \(\chi_{197}(41,\cdot)\) \(\chi_{197}(43,\cdot)\) \(\chi_{197}(47,\cdot)\) \(\chi_{197}(55,\cdot)\) \(\chi_{197}(62,\cdot)\) \(\chi_{197}(64,\cdot)\) \(\chi_{197}(65,\cdot)\) \(\chi_{197}(92,\cdot)\) \(\chi_{197}(96,\cdot)\) \(\chi_{197}(97,\cdot)\) \(\chi_{197}(107,\cdot)\) \(\chi_{197}(109,\cdot)\) \(\chi_{197}(112,\cdot)\) \(\chi_{197}(116,\cdot)\) \(\chi_{197}(121,\cdot)\) \(\chi_{197}(127,\cdot)\) \(\chi_{197}(134,\cdot)\) \(\chi_{197}(136,\cdot)\) \(\chi_{197}(137,\cdot)\) \(\chi_{197}(138,\cdot)\) \(\chi_{197}(143,\cdot)\) ...

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: $\Q(\zeta_{49})$
Fixed field: Number field defined by a degree 98 polynomial

Values on generators

\(2\) → \(e\left(\frac{75}{98}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 197 }(112, a) \) \(1\)\(1\)\(e\left(\frac{75}{98}\right)\)\(e\left(\frac{51}{98}\right)\)\(e\left(\frac{26}{49}\right)\)\(e\left(\frac{11}{98}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{36}{49}\right)\)\(e\left(\frac{29}{98}\right)\)\(e\left(\frac{2}{49}\right)\)\(e\left(\frac{43}{49}\right)\)\(e\left(\frac{19}{98}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 197 }(112,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 197 }(112,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 197 }(112,·),\chi_{ 197 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 197 }(112,·)) \;\) at \(\; a,b = \) e.g. 1,2