Properties

Label 6001.2736
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([22,3]))
 
pari: [g,chi] = znchar(Mod(2736,6001))
 

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.cp

\(\chi_{6001}(135,\cdot)\) \(\chi_{6001}(441,\cdot)\) \(\chi_{6001}(560,\cdot)\) \(\chi_{6001}(1291,\cdot)\) \(\chi_{6001}(1886,\cdot)\) \(\chi_{6001}(2617,\cdot)\) \(\chi_{6001}(2736,\cdot)\) \(\chi_{6001}(3042,\cdot)\) \(\chi_{6001}(3212,\cdot)\) \(\chi_{6001}(3348,\cdot)\) \(\chi_{6001}(4045,\cdot)\) \(\chi_{6001}(4232,\cdot)\) \(\chi_{6001}(4300,\cdot)\) \(\chi_{6001}(4555,\cdot)\) \(\chi_{6001}(4623,\cdot)\) \(\chi_{6001}(4878,\cdot)\) \(\chi_{6001}(4946,\cdot)\) \(\chi_{6001}(5133,\cdot)\) \(\chi_{6001}(5830,\cdot)\) \(\chi_{6001}(5966,\cdot)\)

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

\((2825,3180)\) → \((-1,e\left(\frac{3}{44}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6001 }(2736, a) \) \(1\)\(1\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{25}{44}\right)\)\(-i\)\(i\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(-i\)\(e\left(\frac{13}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6001 }(2736,a) \;\) at \(\;a = \) e.g. 2