Properties

Label 2205.32
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([70,21,8]))
 
pari: [g,chi] = znchar(Mod(32,2205))
 

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.el

\(\chi_{2205}(2,\cdot)\) \(\chi_{2205}(32,\cdot)\) \(\chi_{2205}(158,\cdot)\) \(\chi_{2205}(317,\cdot)\) \(\chi_{2205}(347,\cdot)\) \(\chi_{2205}(443,\cdot)\) \(\chi_{2205}(473,\cdot)\) \(\chi_{2205}(632,\cdot)\) \(\chi_{2205}(662,\cdot)\) \(\chi_{2205}(758,\cdot)\) \(\chi_{2205}(788,\cdot)\) \(\chi_{2205}(947,\cdot)\) \(\chi_{2205}(977,\cdot)\) \(\chi_{2205}(1073,\cdot)\) \(\chi_{2205}(1103,\cdot)\) \(\chi_{2205}(1262,\cdot)\) \(\chi_{2205}(1388,\cdot)\) \(\chi_{2205}(1418,\cdot)\) \(\chi_{2205}(1577,\cdot)\) \(\chi_{2205}(1607,\cdot)\) \(\chi_{2205}(1703,\cdot)\) \(\chi_{2205}(1922,\cdot)\) \(\chi_{2205}(2018,\cdot)\) \(\chi_{2205}(2048,\cdot)\)

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

\((1226,442,1081)\) → \((e\left(\frac{5}{6}\right),i,e\left(\frac{2}{21}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\(1\)\(1\)\(e\left(\frac{47}{84}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{19}{28}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{47}{84}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{11}{84}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{17}{84}\right)\)\(e\left(\frac{15}{28}\right)\)
value at e.g. 2