Properties

Label 3381.22
Modulus $3381$
Conductor $1127$
Order $14$
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(3381, base_ring=CyclotomicField(14))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,8,7]))
 
pari: [g,chi] = znchar(Mod(22,3381))
 

Basic properties

Modulus: \(3381\)
Conductor: \(1127\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(14\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1127}(22,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3381.u

\(\chi_{3381}(22,\cdot)\) \(\chi_{3381}(505,\cdot)\) \(\chi_{3381}(988,\cdot)\) \(\chi_{3381}(1954,\cdot)\) \(\chi_{3381}(2437,\cdot)\) \(\chi_{3381}(2920,\cdot)\)

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: \(\Q(\zeta_{7})\)
Fixed field: 14.0.652300651772147469026072062247.1

Values on generators

\((2255,346,442)\) → \((1,e\left(\frac{4}{7}\right),-1)\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(-1\)\(1\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{11}{14}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(22,a) \;\) at \(\;a = \) e.g. 2