Properties

Label 2001.596
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([77,91,22]))
 
pari: [g,chi] = znchar(Mod(596,2001))
 

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.bn

\(\chi_{2001}(20,\cdot)\) \(\chi_{2001}(53,\cdot)\) \(\chi_{2001}(65,\cdot)\) \(\chi_{2001}(74,\cdot)\) \(\chi_{2001}(83,\cdot)\) \(\chi_{2001}(107,\cdot)\) \(\chi_{2001}(152,\cdot)\) \(\chi_{2001}(194,\cdot)\) \(\chi_{2001}(227,\cdot)\) \(\chi_{2001}(281,\cdot)\) \(\chi_{2001}(314,\cdot)\) \(\chi_{2001}(401,\cdot)\) \(\chi_{2001}(431,\cdot)\) \(\chi_{2001}(458,\cdot)\) \(\chi_{2001}(488,\cdot)\) \(\chi_{2001}(500,\cdot)\) \(\chi_{2001}(596,\cdot)\) \(\chi_{2001}(605,\cdot)\) \(\chi_{2001}(632,\cdot)\) \(\chi_{2001}(674,\cdot)\) \(\chi_{2001}(770,\cdot)\) \(\chi_{2001}(779,\cdot)\) \(\chi_{2001}(803,\cdot)\) \(\chi_{2001}(848,\cdot)\) \(\chi_{2001}(866,\cdot)\) \(\chi_{2001}(893,\cdot)\) \(\chi_{2001}(935,\cdot)\) \(\chi_{2001}(953,\cdot)\) \(\chi_{2001}(977,\cdot)\) \(\chi_{2001}(980,\cdot)\) ...

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

\((668,1132,553)\) → \((-1,e\left(\frac{13}{22}\right),e\left(\frac{1}{7}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{127}{154}\right)\)\(e\left(\frac{50}{77}\right)\)\(e\left(\frac{18}{77}\right)\)\(e\left(\frac{145}{154}\right)\)\(e\left(\frac{73}{154}\right)\)\(e\left(\frac{9}{154}\right)\)\(e\left(\frac{30}{77}\right)\)\(e\left(\frac{65}{77}\right)\)\(e\left(\frac{59}{77}\right)\)\(e\left(\frac{23}{77}\right)\)
value at e.g. 2