Properties

Label 3381.1103
Modulus $3381$
Conductor $3381$
Order $42$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3381, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,16,21]))
 
pari: [g,chi] = znchar(Mod(1103,3381))
 

Basic properties

Modulus: \(3381\)
Conductor: \(3381\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3381.bl

\(\chi_{3381}(137,\cdot)\) \(\chi_{3381}(620,\cdot)\) \(\chi_{3381}(758,\cdot)\) \(\chi_{3381}(1103,\cdot)\) \(\chi_{3381}(1241,\cdot)\) \(\chi_{3381}(1724,\cdot)\) \(\chi_{3381}(2069,\cdot)\) \(\chi_{3381}(2207,\cdot)\) \(\chi_{3381}(2552,\cdot)\) \(\chi_{3381}(2690,\cdot)\) \(\chi_{3381}(3035,\cdot)\) \(\chi_{3381}(3173,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((2255,346,442)\) → \((-1,e\left(\frac{8}{21}\right),-1)\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 3381 }(1103, a) \) \(1\)\(1\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{5}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(1103,a) \;\) at \(\;a = \) e.g. 2