Properties

Label 8033.2029
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([69,124]))
 
pari: [g,chi] = znchar(Mod(2029,8033))
 

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.ca

\(\chi_{8033}(28,\cdot)\) \(\chi_{8033}(57,\cdot)\) \(\chi_{8033}(144,\cdot)\) \(\chi_{8033}(202,\cdot)\) \(\chi_{8033}(492,\cdot)\) \(\chi_{8033}(840,\cdot)\) \(\chi_{8033}(898,\cdot)\) \(\chi_{8033}(985,\cdot)\) \(\chi_{8033}(1072,\cdot)\) \(\chi_{8033}(1101,\cdot)\) \(\chi_{8033}(1623,\cdot)\) \(\chi_{8033}(1710,\cdot)\) \(\chi_{8033}(1942,\cdot)\) \(\chi_{8033}(2029,\cdot)\) \(\chi_{8033}(2406,\cdot)\) \(\chi_{8033}(2464,\cdot)\) \(\chi_{8033}(2841,\cdot)\) \(\chi_{8033}(2870,\cdot)\) \(\chi_{8033}(3102,\cdot)\) \(\chi_{8033}(3218,\cdot)\) \(\chi_{8033}(3334,\cdot)\) \(\chi_{8033}(3624,\cdot)\) \(\chi_{8033}(3682,\cdot)\) \(\chi_{8033}(3856,\cdot)\) \(\chi_{8033}(4204,\cdot)\) \(\chi_{8033}(4291,\cdot)\) \(\chi_{8033}(4320,\cdot)\) \(\chi_{8033}(4349,\cdot)\) \(\chi_{8033}(4407,\cdot)\) \(\chi_{8033}(4523,\cdot)\) ...

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

\((5541,1944)\) → \((-1,e\left(\frac{62}{69}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8033 }(2029, a) \) \(1\)\(1\)\(e\left(\frac{27}{46}\right)\)\(e\left(\frac{59}{138}\right)\)\(e\left(\frac{4}{23}\right)\)\(e\left(\frac{62}{69}\right)\)\(e\left(\frac{1}{69}\right)\)\(e\left(\frac{53}{69}\right)\)\(e\left(\frac{35}{46}\right)\)\(e\left(\frac{59}{69}\right)\)\(e\left(\frac{67}{138}\right)\)\(e\left(\frac{109}{138}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8033 }(2029,a) \;\) at \(\;a = \) e.g. 2