Properties

Label 185.78
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([27,2]))
 
pari: [g,chi] = znchar(Mod(78,185))
 

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.y

\(\chi_{185}(3,\cdot)\) \(\chi_{185}(28,\cdot)\) \(\chi_{185}(58,\cdot)\) \(\chi_{185}(62,\cdot)\) \(\chi_{185}(67,\cdot)\) \(\chi_{185}(77,\cdot)\) \(\chi_{185}(78,\cdot)\) \(\chi_{185}(102,\cdot)\) \(\chi_{185}(132,\cdot)\) \(\chi_{185}(152,\cdot)\) \(\chi_{185}(173,\cdot)\) \(\chi_{185}(178,\cdot)\)

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: 36.0.1552561224397970539319305607385300901659347120825982756912708282470703125.1

Values on generators

\((112,76)\) → \((-i,e\left(\frac{1}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 185 }(78, a) \) \(-1\)\(1\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{11}{18}\right)\)\(-1\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{31}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 185 }(78,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 185 }(78,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 185 }(78,·),\chi_{ 185 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 185 }(78,·)) \;\) at \(\; a,b = \) e.g. 1,2