Properties

Label 8033.637
Modulus $8033$
Conductor $8033$
Order $138$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8033, base_ring=CyclotomicField(138))
 
M = H._module
 
chi = DirichletCharacter(H, M([69,133]))
 
pari: [g,chi] = znchar(Mod(637,8033))
 

Basic properties

Modulus: \(8033\)
Conductor: \(8033\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(138\)
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 8033.bz

\(\chi_{8033}(86,\cdot)\) \(\chi_{8033}(289,\cdot)\) \(\chi_{8033}(347,\cdot)\) \(\chi_{8033}(463,\cdot)\) \(\chi_{8033}(579,\cdot)\) \(\chi_{8033}(637,\cdot)\) \(\chi_{8033}(666,\cdot)\) \(\chi_{8033}(695,\cdot)\) \(\chi_{8033}(782,\cdot)\) \(\chi_{8033}(1130,\cdot)\) \(\chi_{8033}(1304,\cdot)\) \(\chi_{8033}(1362,\cdot)\) \(\chi_{8033}(1652,\cdot)\) \(\chi_{8033}(1768,\cdot)\) \(\chi_{8033}(1884,\cdot)\) \(\chi_{8033}(2116,\cdot)\) \(\chi_{8033}(2145,\cdot)\) \(\chi_{8033}(2522,\cdot)\) \(\chi_{8033}(2580,\cdot)\) \(\chi_{8033}(2957,\cdot)\) \(\chi_{8033}(3044,\cdot)\) \(\chi_{8033}(3276,\cdot)\) \(\chi_{8033}(3363,\cdot)\) \(\chi_{8033}(3885,\cdot)\) \(\chi_{8033}(3914,\cdot)\) \(\chi_{8033}(4001,\cdot)\) \(\chi_{8033}(4088,\cdot)\) \(\chi_{8033}(4146,\cdot)\) \(\chi_{8033}(4494,\cdot)\) \(\chi_{8033}(4784,\cdot)\) ...

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: $\Q(\zeta_{69})$
Fixed field: Number field defined by a degree 138 polynomial (not computed)

Values on generators

\((5541,1944)\) → \((-1,e\left(\frac{133}{138}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8033 }(637, a) \) \(1\)\(1\)\(e\left(\frac{4}{23}\right)\)\(e\left(\frac{95}{138}\right)\)\(e\left(\frac{8}{23}\right)\)\(e\left(\frac{133}{138}\right)\)\(e\left(\frac{119}{138}\right)\)\(e\left(\frac{14}{69}\right)\)\(e\left(\frac{12}{23}\right)\)\(e\left(\frac{26}{69}\right)\)\(e\left(\frac{19}{138}\right)\)\(e\left(\frac{17}{69}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8033 }(637,a) \;\) at \(\;a = \) e.g. 2