Properties

Label 2205.1003
Modulus $2205$
Conductor $2205$
Order $84$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

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([28,63,76]))
 
pari: [g,chi] = znchar(Mod(1003,2205))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2205.ep

\(\chi_{2205}(58,\cdot)\) \(\chi_{2205}(88,\cdot)\) \(\chi_{2205}(247,\cdot)\) \(\chi_{2205}(277,\cdot)\) \(\chi_{2205}(403,\cdot)\) \(\chi_{2205}(562,\cdot)\) \(\chi_{2205}(592,\cdot)\) \(\chi_{2205}(688,\cdot)\) \(\chi_{2205}(718,\cdot)\) \(\chi_{2205}(877,\cdot)\) \(\chi_{2205}(907,\cdot)\) \(\chi_{2205}(1003,\cdot)\) \(\chi_{2205}(1033,\cdot)\) \(\chi_{2205}(1192,\cdot)\) \(\chi_{2205}(1222,\cdot)\) \(\chi_{2205}(1318,\cdot)\) \(\chi_{2205}(1348,\cdot)\) \(\chi_{2205}(1507,\cdot)\) \(\chi_{2205}(1633,\cdot)\) \(\chi_{2205}(1663,\cdot)\) \(\chi_{2205}(1822,\cdot)\) \(\chi_{2205}(1852,\cdot)\) \(\chi_{2205}(1948,\cdot)\) \(\chi_{2205}(2167,\cdot)\)

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

\((1226,442,1081)\) → \((e\left(\frac{1}{3}\right),-i,e\left(\frac{19}{21}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\(-1\)\(1\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{65}{84}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{31}{84}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{84}\right)\)\(e\left(\frac{25}{84}\right)\)
value at e.g. 2