Properties

Label 6001.22
Modulus $6001$
Conductor $6001$
Order $176$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6001, base_ring=CyclotomicField(176))
 
M = H._module
 
chi = DirichletCharacter(H, M([55,112]))
 
pari: [g,chi] = znchar(Mod(22,6001))
 

Basic properties

Modulus: \(6001\)
Conductor: \(6001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(176\)
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 6001.dn

\(\chi_{6001}(22,\cdot)\) \(\chi_{6001}(58,\cdot)\) \(\chi_{6001}(131,\cdot)\) \(\chi_{6001}(231,\cdot)\) \(\chi_{6001}(337,\cdot)\) \(\chi_{6001}(411,\cdot)\) \(\chi_{6001}(538,\cdot)\) \(\chi_{6001}(584,\cdot)\) \(\chi_{6001}(609,\cdot)\) \(\chi_{6001}(690,\cdot)\) \(\chi_{6001}(728,\cdot)\) \(\chi_{6001}(891,\cdot)\) \(\chi_{6001}(923,\cdot)\) \(\chi_{6001}(962,\cdot)\) \(\chi_{6001}(1043,\cdot)\) \(\chi_{6001}(1081,\cdot)\) \(\chi_{6001}(1117,\cdot)\) \(\chi_{6001}(1244,\cdot)\) \(\chi_{6001}(1246,\cdot)\) \(\chi_{6001}(1315,\cdot)\) \(\chi_{6001}(1434,\cdot)\) \(\chi_{6001}(1552,\cdot)\) \(\chi_{6001}(1629,\cdot)\) \(\chi_{6001}(1643,\cdot)\) \(\chi_{6001}(1950,\cdot)\) \(\chi_{6001}(1952,\cdot)\) \(\chi_{6001}(1982,\cdot)\) \(\chi_{6001}(1996,\cdot)\) \(\chi_{6001}(2102,\cdot)\) \(\chi_{6001}(2249,\cdot)\) ...

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: $\Q(\zeta_{176})$
Fixed field: Number field defined by a degree 176 polynomial (not computed)

Values on generators

\((2825,3180)\) → \((e\left(\frac{5}{16}\right),e\left(\frac{7}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6001 }(22, a) \) \(-1\)\(1\)\(e\left(\frac{65}{88}\right)\)\(e\left(\frac{167}{176}\right)\)\(e\left(\frac{21}{44}\right)\)\(e\left(\frac{35}{176}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{19}{88}\right)\)\(e\left(\frac{79}{88}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{65}{176}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6001 }(22,a) \;\) at \(\;a = \) e.g. 2