Properties

Label 2205.68
Modulus $2205$
Conductor $315$
Order $12$
Real no
Primitive no
Minimal no
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(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,9,10]))
 
pari: [g,chi] = znchar(Mod(68,2205))
 

Basic properties

Modulus: \(2205\)
Conductor: \(315\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{315}(68,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2205.bv

\(\chi_{2205}(68,\cdot)\) \(\chi_{2205}(227,\cdot)\) \(\chi_{2205}(668,\cdot)\) \(\chi_{2205}(1832,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.0.213743552925735861328125.2

Values on generators

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

Values

\(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\(-1\)\(1\)\(i\)\(-1\)\(-i\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{12}\right)\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{12}\right)\)
value at e.g. 2