Properties

Label 2205.46
Modulus $2205$
Conductor $49$
Order $21$
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([0,0,22]))
 
pari: [g,chi] = znchar(Mod(46,2205))
 

Basic properties

Modulus: \(2205\)
Conductor: \(49\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{49}(46,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2205.cs

\(\chi_{2205}(46,\cdot)\) \(\chi_{2205}(541,\cdot)\) \(\chi_{2205}(676,\cdot)\) \(\chi_{2205}(856,\cdot)\) \(\chi_{2205}(991,\cdot)\) \(\chi_{2205}(1171,\cdot)\) \(\chi_{2205}(1306,\cdot)\) \(\chi_{2205}(1486,\cdot)\) \(\chi_{2205}(1621,\cdot)\) \(\chi_{2205}(1801,\cdot)\) \(\chi_{2205}(1936,\cdot)\) \(\chi_{2205}(2116,\cdot)\)

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: \(\Q(\zeta_{49})^+\)

Values on generators

\((1226,442,1081)\) → \((1,1,e\left(\frac{11}{21}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\(1\)\(1\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{19}{21}\right)\)
value at e.g. 2