Properties

Label 197.81
Modulus $197$
Conductor $197$
Order $49$
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([68]))
 
pari: [g,chi] = znchar(Mod(81,197))
 

Basic properties

Modulus: \(197\)
Conductor: \(197\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(49\)
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.g

\(\chi_{197}(16,\cdot)\) \(\chi_{197}(23,\cdot)\) \(\chi_{197}(24,\cdot)\) \(\chi_{197}(28,\cdot)\) \(\chi_{197}(29,\cdot)\) \(\chi_{197}(34,\cdot)\) \(\chi_{197}(37,\cdot)\) \(\chi_{197}(40,\cdot)\) \(\chi_{197}(42,\cdot)\) \(\chi_{197}(49,\cdot)\) \(\chi_{197}(51,\cdot)\) \(\chi_{197}(53,\cdot)\) \(\chi_{197}(54,\cdot)\) \(\chi_{197}(59,\cdot)\) \(\chi_{197}(60,\cdot)\) \(\chi_{197}(61,\cdot)\) \(\chi_{197}(63,\cdot)\) \(\chi_{197}(70,\cdot)\) \(\chi_{197}(76,\cdot)\) \(\chi_{197}(81,\cdot)\) \(\chi_{197}(85,\cdot)\) \(\chi_{197}(88,\cdot)\) \(\chi_{197}(90,\cdot)\) \(\chi_{197}(100,\cdot)\) \(\chi_{197}(101,\cdot)\) \(\chi_{197}(105,\cdot)\) \(\chi_{197}(132,\cdot)\) \(\chi_{197}(133,\cdot)\) \(\chi_{197}(135,\cdot)\) \(\chi_{197}(142,\cdot)\) ...

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 49 polynomial

Values on generators

\(2\) → \(e\left(\frac{34}{49}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 197 }(81, a) \) \(1\)\(1\)\(e\left(\frac{34}{49}\right)\)\(e\left(\frac{29}{49}\right)\)\(e\left(\frac{19}{49}\right)\)\(e\left(\frac{37}{49}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{15}{49}\right)\)\(e\left(\frac{4}{49}\right)\)\(e\left(\frac{9}{49}\right)\)\(e\left(\frac{22}{49}\right)\)\(e\left(\frac{6}{49}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 197 }(81,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 197 }(81,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 197 }(81,·),\chi_{ 197 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 197 }(81,·)) \;\) at \(\; a,b = \) e.g. 1,2