Properties

Label 3200.3
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([80,15,56]))
 
pari: [g,chi] = znchar(Mod(3,3200))
 

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.di

\(\chi_{3200}(3,\cdot)\) \(\chi_{3200}(27,\cdot)\) \(\chi_{3200}(83,\cdot)\) \(\chi_{3200}(163,\cdot)\) \(\chi_{3200}(187,\cdot)\) \(\chi_{3200}(267,\cdot)\) \(\chi_{3200}(323,\cdot)\) \(\chi_{3200}(347,\cdot)\) \(\chi_{3200}(403,\cdot)\) \(\chi_{3200}(427,\cdot)\) \(\chi_{3200}(483,\cdot)\) \(\chi_{3200}(563,\cdot)\) \(\chi_{3200}(587,\cdot)\) \(\chi_{3200}(667,\cdot)\) \(\chi_{3200}(723,\cdot)\) \(\chi_{3200}(747,\cdot)\) \(\chi_{3200}(803,\cdot)\) \(\chi_{3200}(827,\cdot)\) \(\chi_{3200}(883,\cdot)\) \(\chi_{3200}(963,\cdot)\) \(\chi_{3200}(987,\cdot)\) \(\chi_{3200}(1067,\cdot)\) \(\chi_{3200}(1123,\cdot)\) \(\chi_{3200}(1147,\cdot)\) \(\chi_{3200}(1203,\cdot)\) \(\chi_{3200}(1227,\cdot)\) \(\chi_{3200}(1283,\cdot)\) \(\chi_{3200}(1363,\cdot)\) \(\chi_{3200}(1387,\cdot)\) \(\chi_{3200}(1467,\cdot)\) ...

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

\((1151,901,2177)\) → \((-1,e\left(\frac{3}{32}\right),e\left(\frac{7}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 3200 }(3, a) \) \(1\)\(1\)\(e\left(\frac{37}{160}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{37}{80}\right)\)\(e\left(\frac{11}{160}\right)\)\(e\left(\frac{9}{160}\right)\)\(e\left(\frac{7}{40}\right)\)\(e\left(\frac{153}{160}\right)\)\(e\left(\frac{67}{160}\right)\)\(e\left(\frac{53}{80}\right)\)\(e\left(\frac{111}{160}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3200 }(3,a) \;\) at \(\;a = \) e.g. 2