Properties

Label 2268.83
Modulus $2268$
Conductor $2268$
Order $54$
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(2268, base_ring=CyclotomicField(54))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([27,1,27]))
 
pari: [g,chi] = znchar(Mod(83,2268))
 

Basic properties

Modulus: \(2268\)
Conductor: \(2268\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(54\)
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 2268.cz

\(\chi_{2268}(83,\cdot)\) \(\chi_{2268}(167,\cdot)\) \(\chi_{2268}(335,\cdot)\) \(\chi_{2268}(419,\cdot)\) \(\chi_{2268}(587,\cdot)\) \(\chi_{2268}(671,\cdot)\) \(\chi_{2268}(839,\cdot)\) \(\chi_{2268}(923,\cdot)\) \(\chi_{2268}(1091,\cdot)\) \(\chi_{2268}(1175,\cdot)\) \(\chi_{2268}(1343,\cdot)\) \(\chi_{2268}(1427,\cdot)\) \(\chi_{2268}(1595,\cdot)\) \(\chi_{2268}(1679,\cdot)\) \(\chi_{2268}(1847,\cdot)\) \(\chi_{2268}(1931,\cdot)\) \(\chi_{2268}(2099,\cdot)\) \(\chi_{2268}(2183,\cdot)\)

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: \(\Q(\zeta_{27})\)
Fixed field: Number field defined by a degree 54 polynomial

Values on generators

\((1135,1541,325)\) → \((-1,e\left(\frac{1}{54}\right),-1)\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\(-1\)\(1\)\(e\left(\frac{25}{27}\right)\)\(e\left(\frac{20}{27}\right)\)\(e\left(\frac{35}{54}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{19}{27}\right)\)\(e\left(\frac{23}{27}\right)\)\(e\left(\frac{37}{54}\right)\)\(e\left(\frac{10}{27}\right)\)\(e\left(\frac{7}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2268 }(83,a) \;\) at \(\;a = \) e.g. 2