Properties

Label 2001.86
Modulus $2001$
Conductor $2001$
Order $22$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2001, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,7,11]))
 
pari: [g,chi] = znchar(Mod(86,2001))
 

Basic properties

Modulus: \(2001\)
Conductor: \(2001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
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 2001.v

\(\chi_{2001}(86,\cdot)\) \(\chi_{2001}(260,\cdot)\) \(\chi_{2001}(434,\cdot)\) \(\chi_{2001}(521,\cdot)\) \(\chi_{2001}(608,\cdot)\) \(\chi_{2001}(695,\cdot)\) \(\chi_{2001}(1217,\cdot)\) \(\chi_{2001}(1391,\cdot)\) \(\chi_{2001}(1652,\cdot)\) \(\chi_{2001}(1739,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((668,1132,553)\) → \((-1,e\left(\frac{7}{22}\right),-1)\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{6}{11}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{11})\)
Fixed field: Number field defined by a degree 22 polynomial