Properties

Label 475.186
Modulus $475$
Conductor $475$
Order $90$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(475, base_ring=CyclotomicField(90))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([72,55]))
 
pari: [g,chi] = znchar(Mod(186,475))
 

Basic properties

Modulus: \(475\)
Conductor: \(475\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
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 475.bf

\(\chi_{475}(21,\cdot)\) \(\chi_{475}(41,\cdot)\) \(\chi_{475}(71,\cdot)\) \(\chi_{475}(86,\cdot)\) \(\chi_{475}(91,\cdot)\) \(\chi_{475}(116,\cdot)\) \(\chi_{475}(136,\cdot)\) \(\chi_{475}(146,\cdot)\) \(\chi_{475}(166,\cdot)\) \(\chi_{475}(181,\cdot)\) \(\chi_{475}(186,\cdot)\) \(\chi_{475}(211,\cdot)\) \(\chi_{475}(231,\cdot)\) \(\chi_{475}(241,\cdot)\) \(\chi_{475}(261,\cdot)\) \(\chi_{475}(281,\cdot)\) \(\chi_{475}(306,\cdot)\) \(\chi_{475}(336,\cdot)\) \(\chi_{475}(356,\cdot)\) \(\chi_{475}(371,\cdot)\) \(\chi_{475}(421,\cdot)\) \(\chi_{475}(431,\cdot)\) \(\chi_{475}(466,\cdot)\) \(\chi_{475}(471,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((77,401)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{11}{18}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\(-1\)\(1\)\(e\left(\frac{37}{90}\right)\)\(e\left(\frac{49}{90}\right)\)\(e\left(\frac{37}{45}\right)\)\(e\left(\frac{43}{45}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{4}{45}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{23}{90}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 475 }(186,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 475 }(186,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 475 }(186,·),\chi_{ 475 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 475 }(186,·)) \;\) at \(\; a,b = \) e.g. 1,2