Properties

Label 4033.20
Modulus $4033$
Conductor $4033$
Order $108$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4033, base_ring=CyclotomicField(108))
 
M = H._module
 
chi = DirichletCharacter(H, M([75,82]))
 
pari: [g,chi] = znchar(Mod(20,4033))
 

Basic properties

Modulus: \(4033\)
Conductor: \(4033\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(108\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4033.iu

\(\chi_{4033}(20,\cdot)\) \(\chi_{4033}(254,\cdot)\) \(\chi_{4033}(279,\cdot)\) \(\chi_{4033}(464,\cdot)\) \(\chi_{4033}(536,\cdot)\) \(\chi_{4033}(557,\cdot)\) \(\chi_{4033}(605,\cdot)\) \(\chi_{4033}(651,\cdot)\) \(\chi_{4033}(685,\cdot)\) \(\chi_{4033}(775,\cdot)\) \(\chi_{4033}(792,\cdot)\) \(\chi_{4033}(1382,\cdot)\) \(\chi_{4033}(1391,\cdot)\) \(\chi_{4033}(1519,\cdot)\) \(\chi_{4033}(1630,\cdot)\) \(\chi_{4033}(1737,\cdot)\) \(\chi_{4033}(1831,\cdot)\) \(\chi_{4033}(1848,\cdot)\) \(\chi_{4033}(1937,\cdot)\) \(\chi_{4033}(2022,\cdot)\) \(\chi_{4033}(2200,\cdot)\) \(\chi_{4033}(2427,\cdot)\) \(\chi_{4033}(2459,\cdot)\) \(\chi_{4033}(2498,\cdot)\) \(\chi_{4033}(2595,\cdot)\) \(\chi_{4033}(2644,\cdot)\) \(\chi_{4033}(2756,\cdot)\) \(\chi_{4033}(2799,\cdot)\) \(\chi_{4033}(2928,\cdot)\) \(\chi_{4033}(3049,\cdot)\) ...

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: $\Q(\zeta_{108})$
Fixed field: Number field defined by a degree 108 polynomial (not computed)

Values on generators

\((1963,2295)\) → \((e\left(\frac{25}{36}\right),e\left(\frac{41}{54}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4033 }(20, a) \) \(-1\)\(1\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{29}{54}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{73}{108}\right)\)\(e\left(\frac{55}{108}\right)\)\(e\left(\frac{16}{27}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{35}{54}\right)\)\(e\left(\frac{23}{27}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4033 }(20,a) \;\) at \(\;a = \) e.g. 2