Properties

Label 2268.47
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([27,7,45]))
 
pari: [g,chi] = znchar(Mod(47,2268))
 

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.da

\(\chi_{2268}(47,\cdot)\) \(\chi_{2268}(59,\cdot)\) \(\chi_{2268}(299,\cdot)\) \(\chi_{2268}(311,\cdot)\) \(\chi_{2268}(551,\cdot)\) \(\chi_{2268}(563,\cdot)\) \(\chi_{2268}(803,\cdot)\) \(\chi_{2268}(815,\cdot)\) \(\chi_{2268}(1055,\cdot)\) \(\chi_{2268}(1067,\cdot)\) \(\chi_{2268}(1307,\cdot)\) \(\chi_{2268}(1319,\cdot)\) \(\chi_{2268}(1559,\cdot)\) \(\chi_{2268}(1571,\cdot)\) \(\chi_{2268}(1811,\cdot)\) \(\chi_{2268}(1823,\cdot)\) \(\chi_{2268}(2063,\cdot)\) \(\chi_{2268}(2075,\cdot)\)

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

\((1135,1541,325)\) → \((-1,e\left(\frac{7}{54}\right),e\left(\frac{5}{6}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\(-1\)\(1\)\(e\left(\frac{4}{27}\right)\)\(e\left(\frac{14}{27}\right)\)\(e\left(\frac{29}{54}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{16}{27}\right)\)\(e\left(\frac{8}{27}\right)\)\(e\left(\frac{43}{54}\right)\)\(e\left(\frac{25}{27}\right)\)\(e\left(\frac{1}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2268 }(47,a) \;\) at \(\;a = \) e.g. 2