Properties

Label 1872.565
Modulus $1872$
Conductor $1872$
Order $12$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1872, base_ring=CyclotomicField(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,3,8,5]))
 
pari: [g,chi] = znchar(Mod(565,1872))
 

Basic properties

Modulus: \(1872\)
Conductor: \(1872\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1872.em

\(\chi_{1872}(565,\cdot)\) \(\chi_{1872}(1237,\cdot)\) \(\chi_{1872}(1597,\cdot)\) \(\chi_{1872}(1645,\cdot)\)

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: 12.0.662684492545134371044048502784.4

Values on generators

\((703,469,209,145)\) → \((1,i,e\left(\frac{2}{3}\right),e\left(\frac{5}{12}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(-1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(-i\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1872 }(565,a) \;\) at \(\;a = \) e.g. 2