Properties

Label 2205.13
Modulus $2205$
Conductor $2205$
Order $84$
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(2205, base_ring=CyclotomicField(84))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([28,63,66]))
 
pari: [g,chi] = znchar(Mod(13,2205))
 

Basic properties

Modulus: \(2205\)
Conductor: \(2205\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
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 2205.ej

\(\chi_{2205}(13,\cdot)\) \(\chi_{2205}(202,\cdot)\) \(\chi_{2205}(223,\cdot)\) \(\chi_{2205}(328,\cdot)\) \(\chi_{2205}(412,\cdot)\) \(\chi_{2205}(517,\cdot)\) \(\chi_{2205}(643,\cdot)\) \(\chi_{2205}(727,\cdot)\) \(\chi_{2205}(853,\cdot)\) \(\chi_{2205}(958,\cdot)\) \(\chi_{2205}(1042,\cdot)\) \(\chi_{2205}(1147,\cdot)\) \(\chi_{2205}(1168,\cdot)\) \(\chi_{2205}(1357,\cdot)\) \(\chi_{2205}(1462,\cdot)\) \(\chi_{2205}(1483,\cdot)\) \(\chi_{2205}(1588,\cdot)\) \(\chi_{2205}(1672,\cdot)\) \(\chi_{2205}(1777,\cdot)\) \(\chi_{2205}(1798,\cdot)\) \(\chi_{2205}(1903,\cdot)\) \(\chi_{2205}(1987,\cdot)\) \(\chi_{2205}(2092,\cdot)\) \(\chi_{2205}(2113,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((1226,442,1081)\) → \((e\left(\frac{1}{3}\right),-i,e\left(\frac{11}{14}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\(1\)\(1\)\(e\left(\frac{43}{84}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{71}{84}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{11}{28}\right)\)\(1\)\(e\left(\frac{23}{84}\right)\)\(e\left(\frac{65}{84}\right)\)
value at e.g. 2