Properties

Label 4560.197
Modulus $4560$
Conductor $4560$
Order $12$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(4560, base_ring=CyclotomicField(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,3,6,3,4]))
 
pari: [g,chi] = znchar(Mod(197,4560))
 

Basic properties

Modulus: \(4560\)
Conductor: \(4560\)
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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4560.fm

\(\chi_{4560}(197,\cdot)\) \(\chi_{4560}(653,\cdot)\) \(\chi_{4560}(3317,\cdot)\) \(\chi_{4560}(3773,\cdot)\)

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: 12.12.207719004173997441024000000000.1

Values on generators

\((1711,1141,3041,2737,1921)\) → \((1,i,-1,i,e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(-i\)\(-i\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(1\)\(-1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4560 }(197,a) \;\) at \(\;a = \) e.g. 2