↑

Properties

Label 475.89
Modulus $475$
Conductor $475$
Order $90$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(475, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([27,25]))
 
Copy content pari:[g,chi] = znchar(Mod(89,475))
 

Basic properties

Modulus: \(475\)
Conductor: \(475\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(90\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 475.bh

\(\chi_{475}(14,\cdot)\) \(\chi_{475}(29,\cdot)\) \(\chi_{475}(34,\cdot)\) \(\chi_{475}(59,\cdot)\) \(\chi_{475}(79,\cdot)\) \(\chi_{475}(89,\cdot)\) \(\chi_{475}(109,\cdot)\) \(\chi_{475}(129,\cdot)\) \(\chi_{475}(154,\cdot)\) \(\chi_{475}(184,\cdot)\) \(\chi_{475}(204,\cdot)\) \(\chi_{475}(219,\cdot)\) \(\chi_{475}(269,\cdot)\) \(\chi_{475}(279,\cdot)\) \(\chi_{475}(314,\cdot)\) \(\chi_{475}(319,\cdot)\) \(\chi_{475}(344,\cdot)\) \(\chi_{475}(364,\cdot)\) \(\chi_{475}(394,\cdot)\) \(\chi_{475}(409,\cdot)\) \(\chi_{475}(414,\cdot)\) \(\chi_{475}(439,\cdot)\) \(\chi_{475}(459,\cdot)\) \(\chi_{475}(469,\cdot)\)

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

Related number fields

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

Values on generators

\((77,401)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{5}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 475 }(89, a) \) \(-1\)\(1\)\(e\left(\frac{26}{45}\right)\)\(e\left(\frac{32}{45}\right)\)\(e\left(\frac{7}{45}\right)\)\(e\left(\frac{13}{45}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{19}{45}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{4}{45}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 475 }(89,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 475 }(89,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 475 }(89,·),\chi_{ 475 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 475 }(89,·)) \;\) at \(\; a,b = \) e.g. 1,2