Properties

Label 2009.197
Modulus $2009$
Conductor $41$
Order $20$
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(2009, base_ring=CyclotomicField(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,9]))
 
pari: [g,chi] = znchar(Mod(197,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(41\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{41}(33,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2009.y

\(\chi_{2009}(197,\cdot)\) \(\chi_{2009}(295,\cdot)\) \(\chi_{2009}(736,\cdot)\) \(\chi_{2009}(1030,\cdot)\) \(\chi_{2009}(1128,\cdot)\) \(\chi_{2009}(1373,\cdot)\) \(\chi_{2009}(1471,\cdot)\) \(\chi_{2009}(1765,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: \(\Q(\zeta_{41})^+\)

Values on generators

\((493,785)\) → \((1,e\left(\frac{9}{20}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{7}{10}\right)\)\(-i\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(-1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{3}{20}\right)\)
value at e.g. 2