Properties

Label 3200.19
Modulus $3200$
Conductor $3200$
Order $160$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3200, base_ring=CyclotomicField(160))
 
M = H._module
 
chi = DirichletCharacter(H, M([80,115,144]))
 
pari: [g,chi] = znchar(Mod(19,3200))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3200.dg

\(\chi_{3200}(19,\cdot)\) \(\chi_{3200}(59,\cdot)\) \(\chi_{3200}(139,\cdot)\) \(\chi_{3200}(179,\cdot)\) \(\chi_{3200}(219,\cdot)\) \(\chi_{3200}(259,\cdot)\) \(\chi_{3200}(339,\cdot)\) \(\chi_{3200}(379,\cdot)\) \(\chi_{3200}(419,\cdot)\) \(\chi_{3200}(459,\cdot)\) \(\chi_{3200}(539,\cdot)\) \(\chi_{3200}(579,\cdot)\) \(\chi_{3200}(619,\cdot)\) \(\chi_{3200}(659,\cdot)\) \(\chi_{3200}(739,\cdot)\) \(\chi_{3200}(779,\cdot)\) \(\chi_{3200}(819,\cdot)\) \(\chi_{3200}(859,\cdot)\) \(\chi_{3200}(939,\cdot)\) \(\chi_{3200}(979,\cdot)\) \(\chi_{3200}(1019,\cdot)\) \(\chi_{3200}(1059,\cdot)\) \(\chi_{3200}(1139,\cdot)\) \(\chi_{3200}(1179,\cdot)\) \(\chi_{3200}(1219,\cdot)\) \(\chi_{3200}(1259,\cdot)\) \(\chi_{3200}(1339,\cdot)\) \(\chi_{3200}(1379,\cdot)\) \(\chi_{3200}(1419,\cdot)\) \(\chi_{3200}(1459,\cdot)\) ...

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

\((1151,901,2177)\) → \((-1,e\left(\frac{23}{32}\right),e\left(\frac{9}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 3200 }(19, a) \) \(-1\)\(1\)\(e\left(\frac{153}{160}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{73}{80}\right)\)\(e\left(\frac{159}{160}\right)\)\(e\left(\frac{141}{160}\right)\)\(e\left(\frac{33}{40}\right)\)\(e\left(\frac{37}{160}\right)\)\(e\left(\frac{23}{160}\right)\)\(e\left(\frac{37}{80}\right)\)\(e\left(\frac{139}{160}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3200 }(19,a) \;\) at \(\;a = \) e.g. 2