Properties

Label 201.128
Modulus $201$
Conductor $201$
Order $66$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([33,7]))
 
pari: [g,chi] = znchar(Mod(128,201))
 

Basic properties

Modulus: \(201\)
Conductor: \(201\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
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 201.p

\(\chi_{201}(2,\cdot)\) \(\chi_{201}(11,\cdot)\) \(\chi_{201}(20,\cdot)\) \(\chi_{201}(32,\cdot)\) \(\chi_{201}(41,\cdot)\) \(\chi_{201}(44,\cdot)\) \(\chi_{201}(50,\cdot)\) \(\chi_{201}(74,\cdot)\) \(\chi_{201}(80,\cdot)\) \(\chi_{201}(95,\cdot)\) \(\chi_{201}(98,\cdot)\) \(\chi_{201}(101,\cdot)\) \(\chi_{201}(113,\cdot)\) \(\chi_{201}(128,\cdot)\) \(\chi_{201}(146,\cdot)\) \(\chi_{201}(152,\cdot)\) \(\chi_{201}(182,\cdot)\) \(\chi_{201}(185,\cdot)\) \(\chi_{201}(191,\cdot)\) \(\chi_{201}(197,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((68,136)\) → \((-1,e\left(\frac{7}{66}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 201 }(128, a) \) \(1\)\(1\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{29}{66}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{1}{66}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{14}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 201 }(128,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 201 }(128,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 201 }(128,·),\chi_{ 201 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 201 }(128,·)) \;\) at \(\; a,b = \) e.g. 1,2