Properties

Label 2268.43
Modulus $2268$
Conductor $324$
Order $54$
Real no
Primitive no
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([27,22,0]))
 
pari: [g,chi] = znchar(Mod(43,2268))
 

Basic properties

Modulus: \(2268\)
Conductor: \(324\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(54\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{324}(43,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2268.cx

\(\chi_{2268}(43,\cdot)\) \(\chi_{2268}(211,\cdot)\) \(\chi_{2268}(295,\cdot)\) \(\chi_{2268}(463,\cdot)\) \(\chi_{2268}(547,\cdot)\) \(\chi_{2268}(715,\cdot)\) \(\chi_{2268}(799,\cdot)\) \(\chi_{2268}(967,\cdot)\) \(\chi_{2268}(1051,\cdot)\) \(\chi_{2268}(1219,\cdot)\) \(\chi_{2268}(1303,\cdot)\) \(\chi_{2268}(1471,\cdot)\) \(\chi_{2268}(1555,\cdot)\) \(\chi_{2268}(1723,\cdot)\) \(\chi_{2268}(1807,\cdot)\) \(\chi_{2268}(1975,\cdot)\) \(\chi_{2268}(2059,\cdot)\) \(\chi_{2268}(2227,\cdot)\)

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

\((1135,1541,325)\) → \((-1,e\left(\frac{11}{27}\right),1)\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\(-1\)\(1\)\(e\left(\frac{10}{27}\right)\)\(e\left(\frac{43}{54}\right)\)\(e\left(\frac{7}{27}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{53}{54}\right)\)\(e\left(\frac{20}{27}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{35}{54}\right)\)\(e\left(\frac{1}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2268 }(43,a) \;\) at \(\;a = \) e.g. 2