Properties

Label 2001.16
Modulus $2001$
Conductor $667$
Order $77$
Real no
Primitive no
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([0,56,22]))
 
pari: [g,chi] = znchar(Mod(16,2001))
 

Basic properties

Modulus: \(2001\)
Conductor: \(667\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(77\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{667}(16,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2001.bk

\(\chi_{2001}(16,\cdot)\) \(\chi_{2001}(25,\cdot)\) \(\chi_{2001}(49,\cdot)\) \(\chi_{2001}(52,\cdot)\) \(\chi_{2001}(82,\cdot)\) \(\chi_{2001}(94,\cdot)\) \(\chi_{2001}(169,\cdot)\) \(\chi_{2001}(190,\cdot)\) \(\chi_{2001}(223,\cdot)\) \(\chi_{2001}(256,\cdot)\) \(\chi_{2001}(397,\cdot)\) \(\chi_{2001}(400,\cdot)\) \(\chi_{2001}(430,\cdot)\) \(\chi_{2001}(487,\cdot)\) \(\chi_{2001}(538,\cdot)\) \(\chi_{2001}(547,\cdot)\) \(\chi_{2001}(604,\cdot)\) \(\chi_{2001}(616,\cdot)\) \(\chi_{2001}(625,\cdot)\) \(\chi_{2001}(634,\cdot)\) \(\chi_{2001}(703,\cdot)\) \(\chi_{2001}(721,\cdot)\) \(\chi_{2001}(745,\cdot)\) \(\chi_{2001}(748,\cdot)\) \(\chi_{2001}(790,\cdot)\) \(\chi_{2001}(808,\cdot)\) \(\chi_{2001}(832,\cdot)\) \(\chi_{2001}(877,\cdot)\) \(\chi_{2001}(886,\cdot)\) \(\chi_{2001}(922,\cdot)\) ...

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

\((668,1132,553)\) → \((1,e\left(\frac{4}{11}\right),e\left(\frac{1}{7}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{67}{77}\right)\)\(e\left(\frac{57}{77}\right)\)\(e\left(\frac{39}{77}\right)\)\(e\left(\frac{48}{77}\right)\)\(e\left(\frac{47}{77}\right)\)\(e\left(\frac{29}{77}\right)\)\(e\left(\frac{65}{77}\right)\)\(e\left(\frac{51}{77}\right)\)\(e\left(\frac{38}{77}\right)\)\(e\left(\frac{37}{77}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{77})$
Fixed field: Number field defined by a degree 77 polynomial