Properties

Label 2205.8
Modulus $2205$
Conductor $735$
Order $28$
Real no
Primitive no
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(28))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([14,21,24]))
 
pari: [g,chi] = znchar(Mod(8,2205))
 

Basic properties

Modulus: \(2205\)
Conductor: \(735\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{735}(8,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2205.cw

\(\chi_{2205}(8,\cdot)\) \(\chi_{2205}(323,\cdot)\) \(\chi_{2205}(512,\cdot)\) \(\chi_{2205}(827,\cdot)\) \(\chi_{2205}(953,\cdot)\) \(\chi_{2205}(1142,\cdot)\) \(\chi_{2205}(1268,\cdot)\) \(\chi_{2205}(1457,\cdot)\) \(\chi_{2205}(1583,\cdot)\) \(\chi_{2205}(1772,\cdot)\) \(\chi_{2205}(1898,\cdot)\) \(\chi_{2205}(2087,\cdot)\)

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: \(\Q(\zeta_{28})\)
Fixed field: 28.28.83709274472667248662717543234670163936409630856037139892578125.1

Values on generators

\((1226,442,1081)\) → \((-1,-i,e\left(\frac{6}{7}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\(1\)\(1\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{19}{28}\right)\)\(-1\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{9}{28}\right)\)
value at e.g. 2