Properties

Label 3200.1473
Modulus $3200$
Conductor $200$
Order $20$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3200, base_ring=CyclotomicField(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,10,11]))
 
pari: [g,chi] = znchar(Mod(1473,3200))
 

Basic properties

Modulus: \(3200\)
Conductor: \(200\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{200}(173,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3200.bz

\(\chi_{3200}(577,\cdot)\) \(\chi_{3200}(833,\cdot)\) \(\chi_{3200}(1217,\cdot)\) \(\chi_{3200}(1473,\cdot)\) \(\chi_{3200}(2113,\cdot)\) \(\chi_{3200}(2497,\cdot)\) \(\chi_{3200}(2753,\cdot)\) \(\chi_{3200}(3137,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: 20.0.3125000000000000000000000000000000.1

Values on generators

\((1151,901,2177)\) → \((1,-1,e\left(\frac{11}{20}\right))\)

Values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 3200 }(1473, a) \) \(-1\)\(1\)\(e\left(\frac{7}{20}\right)\)\(-i\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{1}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3200 }(1473,a) \;\) at \(\;a = \) e.g. 2