Properties

Label 6001.4101
Modulus $6001$
Conductor $6001$
Order $44$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6001, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([33,17]))
 
pari: [g,chi] = znchar(Mod(4101,6001))
 

Basic properties

Modulus: \(6001\)
Conductor: \(6001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
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 6001.ct

\(\chi_{6001}(4,\cdot)\) \(\chi_{6001}(64,\cdot)\) \(\chi_{6001}(319,\cdot)\) \(\chi_{6001}(897,\cdot)\) \(\chi_{6001}(1024,\cdot)\) \(\chi_{6001}(1500,\cdot)\) \(\chi_{6001}(1594,\cdot)\) \(\chi_{6001}(1619,\cdot)\) \(\chi_{6001}(1900,\cdot)\) \(\chi_{6001}(2350,\cdot)\) \(\chi_{6001}(3651,\cdot)\) \(\chi_{6001}(4101,\cdot)\) \(\chi_{6001}(4382,\cdot)\) \(\chi_{6001}(4407,\cdot)\) \(\chi_{6001}(4501,\cdot)\) \(\chi_{6001}(4977,\cdot)\) \(\chi_{6001}(5104,\cdot)\) \(\chi_{6001}(5682,\cdot)\) \(\chi_{6001}(5937,\cdot)\) \(\chi_{6001}(5997,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: Number field defined by a degree 44 polynomial

Values on generators

\((2825,3180)\) → \((-i,e\left(\frac{17}{44}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6001 }(4101, a) \) \(1\)\(1\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(1\)\(-1\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{3}{11}\right)\)\(1\)\(e\left(\frac{19}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6001 }(4101,a) \;\) at \(\;a = \) e.g. 2