Properties

Label 3200.67
Modulus $3200$
Conductor $3200$
Order $160$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3200, base_ring=CyclotomicField(160))
 
M = H._module
 
chi = DirichletCharacter(H, M([80,95,104]))
 
pari: [g,chi] = znchar(Mod(67,3200))
 

Basic properties

Modulus: \(3200\)
Conductor: \(3200\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(160\)
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 3200.dc

\(\chi_{3200}(67,\cdot)\) \(\chi_{3200}(123,\cdot)\) \(\chi_{3200}(147,\cdot)\) \(\chi_{3200}(203,\cdot)\) \(\chi_{3200}(227,\cdot)\) \(\chi_{3200}(283,\cdot)\) \(\chi_{3200}(363,\cdot)\) \(\chi_{3200}(387,\cdot)\) \(\chi_{3200}(467,\cdot)\) \(\chi_{3200}(523,\cdot)\) \(\chi_{3200}(547,\cdot)\) \(\chi_{3200}(603,\cdot)\) \(\chi_{3200}(627,\cdot)\) \(\chi_{3200}(683,\cdot)\) \(\chi_{3200}(763,\cdot)\) \(\chi_{3200}(787,\cdot)\) \(\chi_{3200}(867,\cdot)\) \(\chi_{3200}(923,\cdot)\) \(\chi_{3200}(947,\cdot)\) \(\chi_{3200}(1003,\cdot)\) \(\chi_{3200}(1027,\cdot)\) \(\chi_{3200}(1083,\cdot)\) \(\chi_{3200}(1163,\cdot)\) \(\chi_{3200}(1187,\cdot)\) \(\chi_{3200}(1267,\cdot)\) \(\chi_{3200}(1323,\cdot)\) \(\chi_{3200}(1347,\cdot)\) \(\chi_{3200}(1403,\cdot)\) \(\chi_{3200}(1427,\cdot)\) \(\chi_{3200}(1483,\cdot)\) ...

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

Related number fields

Field of values: $\Q(\zeta_{160})$
Fixed field: Number field defined by a degree 160 polynomial (not computed)

Values on generators

\((1151,901,2177)\) → \((-1,e\left(\frac{19}{32}\right),e\left(\frac{13}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 3200 }(67, a) \) \(1\)\(1\)\(e\left(\frac{133}{160}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{53}{80}\right)\)\(e\left(\frac{59}{160}\right)\)\(e\left(\frac{41}{160}\right)\)\(e\left(\frac{3}{40}\right)\)\(e\left(\frac{137}{160}\right)\)\(e\left(\frac{83}{160}\right)\)\(e\left(\frac{77}{80}\right)\)\(e\left(\frac{79}{160}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3200 }(67,a) \;\) at \(\;a = \) e.g. 2