Properties

Label 2268.67
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([27,44,36]))
 
pari: [g,chi] = znchar(Mod(67,2268))
 

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.cy

\(\chi_{2268}(67,\cdot)\) \(\chi_{2268}(79,\cdot)\) \(\chi_{2268}(319,\cdot)\) \(\chi_{2268}(331,\cdot)\) \(\chi_{2268}(571,\cdot)\) \(\chi_{2268}(583,\cdot)\) \(\chi_{2268}(823,\cdot)\) \(\chi_{2268}(835,\cdot)\) \(\chi_{2268}(1075,\cdot)\) \(\chi_{2268}(1087,\cdot)\) \(\chi_{2268}(1327,\cdot)\) \(\chi_{2268}(1339,\cdot)\) \(\chi_{2268}(1579,\cdot)\) \(\chi_{2268}(1591,\cdot)\) \(\chi_{2268}(1831,\cdot)\) \(\chi_{2268}(1843,\cdot)\) \(\chi_{2268}(2083,\cdot)\) \(\chi_{2268}(2095,\cdot)\)

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

\((1135,1541,325)\) → \((-1,e\left(\frac{22}{27}\right),e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\(-1\)\(1\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{41}{54}\right)\)\(e\left(\frac{14}{27}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{43}{54}\right)\)\(e\left(\frac{4}{27}\right)\)\(e\left(\frac{4}{27}\right)\)\(e\left(\frac{25}{54}\right)\)\(e\left(\frac{5}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2268 }(67,a) \;\) at \(\;a = \) e.g. 2