Properties

Label 475.291
Modulus $475$
Conductor $475$
Order $45$
Real no
Primitive yes
Minimal yes
Parity even

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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([18,70]))
 
pari: [g,chi] = znchar(Mod(291,475))
 

Basic properties

Modulus: \(475\)
Conductor: \(475\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(45\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 475.bc

\(\chi_{475}(6,\cdot)\) \(\chi_{475}(16,\cdot)\) \(\chi_{475}(36,\cdot)\) \(\chi_{475}(61,\cdot)\) \(\chi_{475}(66,\cdot)\) \(\chi_{475}(81,\cdot)\) \(\chi_{475}(111,\cdot)\) \(\chi_{475}(131,\cdot)\) \(\chi_{475}(156,\cdot)\) \(\chi_{475}(161,\cdot)\) \(\chi_{475}(196,\cdot)\) \(\chi_{475}(206,\cdot)\) \(\chi_{475}(256,\cdot)\) \(\chi_{475}(271,\cdot)\) \(\chi_{475}(291,\cdot)\) \(\chi_{475}(321,\cdot)\) \(\chi_{475}(346,\cdot)\) \(\chi_{475}(366,\cdot)\) \(\chi_{475}(386,\cdot)\) \(\chi_{475}(396,\cdot)\) \(\chi_{475}(416,\cdot)\) \(\chi_{475}(441,\cdot)\) \(\chi_{475}(446,\cdot)\) \(\chi_{475}(461,\cdot)\)

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: 45.45.299215681303998835585125432825671967739342947202402933846152911778748517690473818220198154449462890625.1

Values on generators

\((77,401)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{7}{9}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 475 }(291, a) \) \(1\)\(1\)\(e\left(\frac{44}{45}\right)\)\(e\left(\frac{23}{45}\right)\)\(e\left(\frac{43}{45}\right)\)\(e\left(\frac{22}{45}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{1}{45}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{31}{45}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 475 }(291,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 475 }(291,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 475 }(291,·),\chi_{ 475 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 475 }(291,·)) \;\) at \(\; a,b = \) e.g. 1,2