Properties

Label 2001.410
Modulus $2001$
Conductor $2001$
Order $154$
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(154))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([77,105,11]))
 
pari: [g,chi] = znchar(Mod(410,2001))
 

Basic properties

Modulus: \(2001\)
Conductor: \(2001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(154\)
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.br

\(\chi_{2001}(5,\cdot)\) \(\chi_{2001}(38,\cdot)\) \(\chi_{2001}(80,\cdot)\) \(\chi_{2001}(122,\cdot)\) \(\chi_{2001}(125,\cdot)\) \(\chi_{2001}(149,\cdot)\) \(\chi_{2001}(158,\cdot)\) \(\chi_{2001}(212,\cdot)\) \(\chi_{2001}(245,\cdot)\) \(\chi_{2001}(296,\cdot)\) \(\chi_{2001}(332,\cdot)\) \(\chi_{2001}(341,\cdot)\) \(\chi_{2001}(383,\cdot)\) \(\chi_{2001}(410,\cdot)\) \(\chi_{2001}(419,\cdot)\) \(\chi_{2001}(428,\cdot)\) \(\chi_{2001}(470,\cdot)\) \(\chi_{2001}(497,\cdot)\) \(\chi_{2001}(527,\cdot)\) \(\chi_{2001}(557,\cdot)\) \(\chi_{2001}(701,\cdot)\) \(\chi_{2001}(734,\cdot)\) \(\chi_{2001}(776,\cdot)\) \(\chi_{2001}(845,\cdot)\) \(\chi_{2001}(908,\cdot)\) \(\chi_{2001}(941,\cdot)\) \(\chi_{2001}(950,\cdot)\) \(\chi_{2001}(962,\cdot)\) \(\chi_{2001}(1019,\cdot)\) \(\chi_{2001}(1049,\cdot)\) ...

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

Related number fields

Field of values: $\Q(\zeta_{77})$
Fixed field: Number field defined by a degree 154 polynomial (not computed)

Values on generators

\((668,1132,553)\) → \((-1,e\left(\frac{15}{22}\right),e\left(\frac{1}{14}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{72}{77}\right)\)\(e\left(\frac{67}{77}\right)\)\(e\left(\frac{58}{77}\right)\)\(e\left(\frac{125}{154}\right)\)\(e\left(\frac{62}{77}\right)\)\(e\left(\frac{53}{77}\right)\)\(e\left(\frac{65}{154}\right)\)\(e\left(\frac{64}{77}\right)\)\(e\left(\frac{115}{154}\right)\)\(e\left(\frac{57}{77}\right)\)
value at e.g. 2