Properties

Label 2205.47
Modulus $2205$
Conductor $2205$
Order $84$
Real no
Primitive yes
Minimal yes
Parity odd

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([14,21,10]))
 
pari: [g,chi] = znchar(Mod(47,2205))
 

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.ek

\(\chi_{2205}(47,\cdot)\) \(\chi_{2205}(122,\cdot)\) \(\chi_{2205}(173,\cdot)\) \(\chi_{2205}(248,\cdot)\) \(\chi_{2205}(437,\cdot)\) \(\chi_{2205}(488,\cdot)\) \(\chi_{2205}(563,\cdot)\) \(\chi_{2205}(677,\cdot)\) \(\chi_{2205}(752,\cdot)\) \(\chi_{2205}(878,\cdot)\) \(\chi_{2205}(992,\cdot)\) \(\chi_{2205}(1067,\cdot)\) \(\chi_{2205}(1118,\cdot)\) \(\chi_{2205}(1193,\cdot)\) \(\chi_{2205}(1307,\cdot)\) \(\chi_{2205}(1382,\cdot)\) \(\chi_{2205}(1433,\cdot)\) \(\chi_{2205}(1508,\cdot)\) \(\chi_{2205}(1622,\cdot)\) \(\chi_{2205}(1748,\cdot)\) \(\chi_{2205}(1823,\cdot)\) \(\chi_{2205}(1937,\cdot)\) \(\chi_{2205}(2012,\cdot)\) \(\chi_{2205}(2063,\cdot)\)

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

\((1226,442,1081)\) → \((e\left(\frac{1}{6}\right),i,e\left(\frac{5}{42}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\(-1\)\(1\)\(e\left(\frac{43}{84}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{1}{84}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{61}{84}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{37}{84}\right)\)\(e\left(\frac{3}{28}\right)\)
value at e.g. 2