Properties

Label 8027.1818
Modulus $8027$
Conductor $349$
Order $174$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8027, base_ring=CyclotomicField(174))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,19]))
 
pari: [g,chi] = znchar(Mod(1818,8027))
 

Basic properties

Modulus: \(8027\)
Conductor: \(349\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(174\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{349}(73,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8027.bg

\(\chi_{8027}(70,\cdot)\) \(\chi_{8027}(93,\cdot)\) \(\chi_{8027}(323,\cdot)\) \(\chi_{8027}(553,\cdot)\) \(\chi_{8027}(1151,\cdot)\) \(\chi_{8027}(1381,\cdot)\) \(\chi_{8027}(1565,\cdot)\) \(\chi_{8027}(1634,\cdot)\) \(\chi_{8027}(1726,\cdot)\) \(\chi_{8027}(1749,\cdot)\) \(\chi_{8027}(1818,\cdot)\) \(\chi_{8027}(1887,\cdot)\) \(\chi_{8027}(2071,\cdot)\) \(\chi_{8027}(2485,\cdot)\) \(\chi_{8027}(2600,\cdot)\) \(\chi_{8027}(2715,\cdot)\) \(\chi_{8027}(2922,\cdot)\) \(\chi_{8027}(2945,\cdot)\) \(\chi_{8027}(3060,\cdot)\) \(\chi_{8027}(3129,\cdot)\) \(\chi_{8027}(3198,\cdot)\) \(\chi_{8027}(3382,\cdot)\) \(\chi_{8027}(3405,\cdot)\) \(\chi_{8027}(3474,\cdot)\) \(\chi_{8027}(3566,\cdot)\) \(\chi_{8027}(3681,\cdot)\) \(\chi_{8027}(3704,\cdot)\) \(\chi_{8027}(3819,\cdot)\) \(\chi_{8027}(3842,\cdot)\) \(\chi_{8027}(3888,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{87})$
Fixed field: Number field defined by a degree 174 polynomial (not computed)

Values on generators

\((350,5935)\) → \((1,e\left(\frac{19}{174}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8027 }(1818, a) \) \(1\)\(1\)\(e\left(\frac{19}{174}\right)\)\(e\left(\frac{73}{87}\right)\)\(e\left(\frac{19}{87}\right)\)\(e\left(\frac{25}{87}\right)\)\(e\left(\frac{55}{58}\right)\)\(e\left(\frac{55}{174}\right)\)\(e\left(\frac{19}{58}\right)\)\(e\left(\frac{59}{87}\right)\)\(e\left(\frac{23}{58}\right)\)\(e\left(\frac{17}{58}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8027 }(1818,a) \;\) at \(\;a = \) e.g. 2