Properties

Label 475.17
Modulus $475$
Conductor $475$
Order $180$
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(180))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([117,100]))
 
pari: [g,chi] = znchar(Mod(17,475))
 

Basic properties

Modulus: \(475\)
Conductor: \(475\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(180\)
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.bj

\(\chi_{475}(17,\cdot)\) \(\chi_{475}(23,\cdot)\) \(\chi_{475}(28,\cdot)\) \(\chi_{475}(42,\cdot)\) \(\chi_{475}(47,\cdot)\) \(\chi_{475}(62,\cdot)\) \(\chi_{475}(63,\cdot)\) \(\chi_{475}(73,\cdot)\) \(\chi_{475}(92,\cdot)\) \(\chi_{475}(112,\cdot)\) \(\chi_{475}(123,\cdot)\) \(\chi_{475}(137,\cdot)\) \(\chi_{475}(138,\cdot)\) \(\chi_{475}(142,\cdot)\) \(\chi_{475}(158,\cdot)\) \(\chi_{475}(177,\cdot)\) \(\chi_{475}(187,\cdot)\) \(\chi_{475}(188,\cdot)\) \(\chi_{475}(213,\cdot)\) \(\chi_{475}(233,\cdot)\) \(\chi_{475}(237,\cdot)\) \(\chi_{475}(252,\cdot)\) \(\chi_{475}(253,\cdot)\) \(\chi_{475}(263,\cdot)\) \(\chi_{475}(272,\cdot)\) \(\chi_{475}(283,\cdot)\) \(\chi_{475}(302,\cdot)\) \(\chi_{475}(308,\cdot)\) \(\chi_{475}(313,\cdot)\) \(\chi_{475}(327,\cdot)\) ...

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: $\Q(\zeta_{180})$
Fixed field: Number field defined by a degree 180 polynomial (not computed)

Values on generators

\((77,401)\) → \((e\left(\frac{13}{20}\right),e\left(\frac{5}{9}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\(-1\)\(1\)\(e\left(\frac{37}{180}\right)\)\(e\left(\frac{139}{180}\right)\)\(e\left(\frac{37}{90}\right)\)\(e\left(\frac{44}{45}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{49}{90}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{23}{180}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 475 }(17,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 475 }(17,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 475 }(17,·),\chi_{ 475 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 475 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2