Properties

Label 2001.200
Modulus $2001$
Conductor $2001$
Order $308$
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(308))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([154,112,209]))
 
pari: [g,chi] = znchar(Mod(200,2001))
 

Basic properties

Modulus: \(2001\)
Conductor: \(2001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(308\)
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.bv

\(\chi_{2001}(2,\cdot)\) \(\chi_{2001}(8,\cdot)\) \(\chi_{2001}(26,\cdot)\) \(\chi_{2001}(32,\cdot)\) \(\chi_{2001}(50,\cdot)\) \(\chi_{2001}(77,\cdot)\) \(\chi_{2001}(95,\cdot)\) \(\chi_{2001}(98,\cdot)\) \(\chi_{2001}(101,\cdot)\) \(\chi_{2001}(119,\cdot)\) \(\chi_{2001}(131,\cdot)\) \(\chi_{2001}(164,\cdot)\) \(\chi_{2001}(188,\cdot)\) \(\chi_{2001}(200,\cdot)\) \(\chi_{2001}(242,\cdot)\) \(\chi_{2001}(269,\cdot)\) \(\chi_{2001}(305,\cdot)\) \(\chi_{2001}(308,\cdot)\) \(\chi_{2001}(311,\cdot)\) \(\chi_{2001}(317,\cdot)\) \(\chi_{2001}(338,\cdot)\) \(\chi_{2001}(374,\cdot)\) \(\chi_{2001}(380,\cdot)\) \(\chi_{2001}(395,\cdot)\) \(\chi_{2001}(404,\cdot)\) \(\chi_{2001}(416,\cdot)\) \(\chi_{2001}(443,\cdot)\) \(\chi_{2001}(446,\cdot)\) \(\chi_{2001}(449,\cdot)\) \(\chi_{2001}(485,\cdot)\) ...

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: $\Q(\zeta_{308})$
Fixed field: Number field defined by a degree 308 polynomial (not computed)

Values on generators

\((668,1132,553)\) → \((-1,e\left(\frac{4}{11}\right),e\left(\frac{19}{28}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{279}{308}\right)\)\(e\left(\frac{125}{154}\right)\)\(e\left(\frac{61}{77}\right)\)\(e\left(\frac{4}{77}\right)\)\(e\left(\frac{221}{308}\right)\)\(e\left(\frac{215}{308}\right)\)\(e\left(\frac{227}{308}\right)\)\(e\left(\frac{47}{154}\right)\)\(e\left(\frac{295}{308}\right)\)\(e\left(\frac{48}{77}\right)\)
value at e.g. 2