Properties

Label 2205.794
Modulus $2205$
Conductor $2205$
Order $42$
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(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([7,21,13]))
 
pari: [g,chi] = znchar(Mod(794,2205))
 

Basic properties

Modulus: \(2205\)
Conductor: \(2205\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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.cz

\(\chi_{2205}(164,\cdot)\) \(\chi_{2205}(194,\cdot)\) \(\chi_{2205}(479,\cdot)\) \(\chi_{2205}(794,\cdot)\) \(\chi_{2205}(824,\cdot)\) \(\chi_{2205}(1139,\cdot)\) \(\chi_{2205}(1424,\cdot)\) \(\chi_{2205}(1454,\cdot)\) \(\chi_{2205}(1739,\cdot)\) \(\chi_{2205}(1769,\cdot)\) \(\chi_{2205}(2054,\cdot)\) \(\chi_{2205}(2084,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: 42.42.64499777946714835177141019992254259402911208109553981749879955329445342864388015575823925406164646148681640625.1

Values on generators

\((1226,442,1081)\) → \((e\left(\frac{1}{6}\right),-1,e\left(\frac{13}{42}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\(1\)\(1\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{2}{21}\right)\)
value at e.g. 2