Properties

Label 2268.131
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([27,47,45]))
 
pari: [g,chi] = znchar(Mod(131,2268))
 

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.cq

\(\chi_{2268}(131,\cdot)\) \(\chi_{2268}(227,\cdot)\) \(\chi_{2268}(383,\cdot)\) \(\chi_{2268}(479,\cdot)\) \(\chi_{2268}(635,\cdot)\) \(\chi_{2268}(731,\cdot)\) \(\chi_{2268}(887,\cdot)\) \(\chi_{2268}(983,\cdot)\) \(\chi_{2268}(1139,\cdot)\) \(\chi_{2268}(1235,\cdot)\) \(\chi_{2268}(1391,\cdot)\) \(\chi_{2268}(1487,\cdot)\) \(\chi_{2268}(1643,\cdot)\) \(\chi_{2268}(1739,\cdot)\) \(\chi_{2268}(1895,\cdot)\) \(\chi_{2268}(1991,\cdot)\) \(\chi_{2268}(2147,\cdot)\) \(\chi_{2268}(2243,\cdot)\)

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

\((1135,1541,325)\) → \((-1,e\left(\frac{47}{54}\right),e\left(\frac{5}{6}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\(-1\)\(1\)\(e\left(\frac{5}{27}\right)\)\(e\left(\frac{4}{27}\right)\)\(e\left(\frac{25}{54}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{20}{27}\right)\)\(e\left(\frac{10}{27}\right)\)\(e\left(\frac{11}{54}\right)\)\(e\left(\frac{20}{27}\right)\)\(e\left(\frac{2}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2268 }(131,a) \;\) at \(\;a = \) e.g. 2