Properties

Label 2205.26
Modulus $2205$
Conductor $147$
Order $42$
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(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,0,17]))
 
pari: [g,chi] = znchar(Mod(26,2205))
 

Basic properties

Modulus: \(2205\)
Conductor: \(147\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{147}(26,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2205.dg

\(\chi_{2205}(26,\cdot)\) \(\chi_{2205}(206,\cdot)\) \(\chi_{2205}(341,\cdot)\) \(\chi_{2205}(836,\cdot)\) \(\chi_{2205}(971,\cdot)\) \(\chi_{2205}(1151,\cdot)\) \(\chi_{2205}(1286,\cdot)\) \(\chi_{2205}(1466,\cdot)\) \(\chi_{2205}(1601,\cdot)\) \(\chi_{2205}(1781,\cdot)\) \(\chi_{2205}(1916,\cdot)\) \(\chi_{2205}(2096,\cdot)\)

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: \(\Q(\zeta_{147})^+\)

Values on generators

\((1226,442,1081)\) → \((-1,1,e\left(\frac{17}{42}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\(1\)\(1\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{37}{42}\right)\)
value at e.g. 2