Properties

Label 2205.61
Modulus $2205$
Conductor $441$
Order $42$
Real no
Primitive no
Minimal yes
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(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([28,0,11]))
 
pari: [g,chi] = znchar(Mod(61,2205))
 

Basic properties

Modulus: \(2205\)
Conductor: \(441\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{441}(61,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2205.dn

\(\chi_{2205}(61,\cdot)\) \(\chi_{2205}(346,\cdot)\) \(\chi_{2205}(376,\cdot)\) \(\chi_{2205}(661,\cdot)\) \(\chi_{2205}(691,\cdot)\) \(\chi_{2205}(976,\cdot)\) \(\chi_{2205}(1006,\cdot)\) \(\chi_{2205}(1291,\cdot)\) \(\chi_{2205}(1321,\cdot)\) \(\chi_{2205}(1606,\cdot)\) \(\chi_{2205}(1921,\cdot)\) \(\chi_{2205}(1951,\cdot)\)

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: 42.0.61849948934846323740928964041516234392013738413062346563659921389600804608476019954673203847.2

Values on generators

\((1226,442,1081)\) → \((e\left(\frac{2}{3}\right),1,e\left(\frac{11}{42}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\(-1\)\(1\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{2}{7}\right)\)
value at e.g. 2