Properties

Label 3381.50
Modulus $3381$
Conductor $69$
Order $22$
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(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,0,4]))
 
pari: [g,chi] = znchar(Mod(50,3381))
 

Basic properties

Modulus: \(3381\)
Conductor: \(69\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{69}(50,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3381.bg

\(\chi_{3381}(50,\cdot)\) \(\chi_{3381}(197,\cdot)\) \(\chi_{3381}(491,\cdot)\) \(\chi_{3381}(785,\cdot)\) \(\chi_{3381}(932,\cdot)\) \(\chi_{3381}(1373,\cdot)\) \(\chi_{3381}(1520,\cdot)\) \(\chi_{3381}(1961,\cdot)\) \(\chi_{3381}(3137,\cdot)\) \(\chi_{3381}(3284,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: 22.0.304011857053427966889939263171547.1

Values on generators

\((2255,346,442)\) → \((-1,1,e\left(\frac{2}{11}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(-1\)\(1\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{8}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(50,a) \;\) at \(\;a = \) e.g. 2