Properties

Label 2205.52
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([56,21,2]))
 
pari: [g,chi] = znchar(Mod(52,2205))
 

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.eo

\(\chi_{2205}(52,\cdot)\) \(\chi_{2205}(103,\cdot)\) \(\chi_{2205}(292,\cdot)\) \(\chi_{2205}(367,\cdot)\) \(\chi_{2205}(418,\cdot)\) \(\chi_{2205}(493,\cdot)\) \(\chi_{2205}(682,\cdot)\) \(\chi_{2205}(733,\cdot)\) \(\chi_{2205}(808,\cdot)\) \(\chi_{2205}(922,\cdot)\) \(\chi_{2205}(997,\cdot)\) \(\chi_{2205}(1123,\cdot)\) \(\chi_{2205}(1237,\cdot)\) \(\chi_{2205}(1312,\cdot)\) \(\chi_{2205}(1363,\cdot)\) \(\chi_{2205}(1438,\cdot)\) \(\chi_{2205}(1552,\cdot)\) \(\chi_{2205}(1627,\cdot)\) \(\chi_{2205}(1678,\cdot)\) \(\chi_{2205}(1753,\cdot)\) \(\chi_{2205}(1867,\cdot)\) \(\chi_{2205}(1993,\cdot)\) \(\chi_{2205}(2068,\cdot)\) \(\chi_{2205}(2182,\cdot)\)

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

\((1226,442,1081)\) → \((e\left(\frac{2}{3}\right),i,e\left(\frac{1}{42}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\(1\)\(1\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{73}{84}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{71}{84}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{13}{84}\right)\)\(e\left(\frac{83}{84}\right)\)
value at e.g. 2