Properties

Label 3200.23
Modulus $3200$
Conductor $1600$
Order $80$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3200, base_ring=CyclotomicField(80))
 
M = H._module
 
chi = DirichletCharacter(H, M([40,35,44]))
 
pari: [g,chi] = znchar(Mod(23,3200))
 

Basic properties

Modulus: \(3200\)
Conductor: \(1600\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(80\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1600}(723,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3200.cu

\(\chi_{3200}(23,\cdot)\) \(\chi_{3200}(167,\cdot)\) \(\chi_{3200}(183,\cdot)\) \(\chi_{3200}(327,\cdot)\) \(\chi_{3200}(487,\cdot)\) \(\chi_{3200}(503,\cdot)\) \(\chi_{3200}(647,\cdot)\) \(\chi_{3200}(663,\cdot)\) \(\chi_{3200}(823,\cdot)\) \(\chi_{3200}(967,\cdot)\) \(\chi_{3200}(983,\cdot)\) \(\chi_{3200}(1127,\cdot)\) \(\chi_{3200}(1287,\cdot)\) \(\chi_{3200}(1303,\cdot)\) \(\chi_{3200}(1447,\cdot)\) \(\chi_{3200}(1463,\cdot)\) \(\chi_{3200}(1623,\cdot)\) \(\chi_{3200}(1767,\cdot)\) \(\chi_{3200}(1783,\cdot)\) \(\chi_{3200}(1927,\cdot)\) \(\chi_{3200}(2087,\cdot)\) \(\chi_{3200}(2103,\cdot)\) \(\chi_{3200}(2247,\cdot)\) \(\chi_{3200}(2263,\cdot)\) \(\chi_{3200}(2423,\cdot)\) \(\chi_{3200}(2567,\cdot)\) \(\chi_{3200}(2583,\cdot)\) \(\chi_{3200}(2727,\cdot)\) \(\chi_{3200}(2887,\cdot)\) \(\chi_{3200}(2903,\cdot)\) ...

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: $\Q(\zeta_{80})$
Fixed field: Number field defined by a degree 80 polynomial

Values on generators

\((1151,901,2177)\) → \((-1,e\left(\frac{7}{16}\right),e\left(\frac{11}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 3200 }(23, a) \) \(1\)\(1\)\(e\left(\frac{53}{80}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{13}{40}\right)\)\(e\left(\frac{39}{80}\right)\)\(e\left(\frac{1}{80}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{37}{80}\right)\)\(e\left(\frac{23}{80}\right)\)\(e\left(\frac{27}{40}\right)\)\(e\left(\frac{79}{80}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3200 }(23,a) \;\) at \(\;a = \) e.g. 2