Properties

Label 2001.1409
Modulus $2001$
Conductor $2001$
Order $44$
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(44))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([22,36,33]))
 
pari: [g,chi] = znchar(Mod(1409,2001))
 

Basic properties

Modulus: \(2001\)
Conductor: \(2001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
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.bg

\(\chi_{2001}(41,\cdot)\) \(\chi_{2001}(104,\cdot)\) \(\chi_{2001}(128,\cdot)\) \(\chi_{2001}(215,\cdot)\) \(\chi_{2001}(278,\cdot)\) \(\chi_{2001}(302,\cdot)\) \(\chi_{2001}(476,\cdot)\) \(\chi_{2001}(650,\cdot)\) \(\chi_{2001}(800,\cdot)\) \(\chi_{2001}(887,\cdot)\) \(\chi_{2001}(974,\cdot)\) \(\chi_{2001}(998,\cdot)\) \(\chi_{2001}(1061,\cdot)\) \(\chi_{2001}(1085,\cdot)\) \(\chi_{2001}(1235,\cdot)\) \(\chi_{2001}(1346,\cdot)\) \(\chi_{2001}(1409,\cdot)\) \(\chi_{2001}(1520,\cdot)\) \(\chi_{2001}(1757,\cdot)\) \(\chi_{2001}(1844,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: Number field defined by a degree 44 polynomial

Values on generators

\((668,1132,553)\) → \((-1,e\left(\frac{9}{11}\right),-i)\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{39}{44}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{29}{44}\right)\)\(e\left(\frac{31}{44}\right)\)\(e\left(\frac{27}{44}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{19}{44}\right)\)\(e\left(\frac{6}{11}\right)\)
value at e.g. 2