Properties

Label 185.118
Modulus $185$
Conductor $185$
Order $36$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(185, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([27,32]))
 
pari: [g,chi] = znchar(Mod(118,185))
 

Basic properties

Modulus: \(185\)
Conductor: \(185\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
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 185.bd

\(\chi_{185}(7,\cdot)\) \(\chi_{185}(12,\cdot)\) \(\chi_{185}(33,\cdot)\) \(\chi_{185}(53,\cdot)\) \(\chi_{185}(83,\cdot)\) \(\chi_{185}(107,\cdot)\) \(\chi_{185}(108,\cdot)\) \(\chi_{185}(118,\cdot)\) \(\chi_{185}(123,\cdot)\) \(\chi_{185}(127,\cdot)\) \(\chi_{185}(157,\cdot)\) \(\chi_{185}(182,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: 36.0.1134084166835624937413663701523229292665702790961273014545440673828125.1

Values on generators

\((112,76)\) → \((-i,e\left(\frac{8}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 185 }(118, a) \) \(-1\)\(1\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{13}{36}\right)\)\(e\left(\frac{5}{18}\right)\)\(1\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{1}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 185 }(118,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 185 }(118,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 185 }(118,·),\chi_{ 185 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 185 }(118,·)) \;\) at \(\; a,b = \) e.g. 1,2