Properties

Label 4033.1271
Modulus $4033$
Conductor $4033$
Order $108$
Real no
Primitive yes
Minimal yes
Parity even

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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([33,59]))
 
pari: [g,chi] = znchar(Mod(1271,4033))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4033.if

\(\chi_{4033}(52,\cdot)\) \(\chi_{4033}(165,\cdot)\) \(\chi_{4033}(205,\cdot)\) \(\chi_{4033}(357,\cdot)\) \(\chi_{4033}(425,\cdot)\) \(\chi_{4033}(446,\cdot)\) \(\chi_{4033}(668,\cdot)\) \(\chi_{4033}(698,\cdot)\) \(\chi_{4033}(723,\cdot)\) \(\chi_{4033}(745,\cdot)\) \(\chi_{4033}(1023,\cdot)\) \(\chi_{4033}(1129,\cdot)\) \(\chi_{4033}(1152,\cdot)\) \(\chi_{4033}(1271,\cdot)\) \(\chi_{4033}(1441,\cdot)\) \(\chi_{4033}(1532,\cdot)\) \(\chi_{4033}(1794,\cdot)\) \(\chi_{4033}(2013,\cdot)\) \(\chi_{4033}(2020,\cdot)\) \(\chi_{4033}(2239,\cdot)\) \(\chi_{4033}(2501,\cdot)\) \(\chi_{4033}(2592,\cdot)\) \(\chi_{4033}(2762,\cdot)\) \(\chi_{4033}(2881,\cdot)\) \(\chi_{4033}(2904,\cdot)\) \(\chi_{4033}(3010,\cdot)\) \(\chi_{4033}(3288,\cdot)\) \(\chi_{4033}(3310,\cdot)\) \(\chi_{4033}(3335,\cdot)\) \(\chi_{4033}(3365,\cdot)\) ...

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

\((1963,2295)\) → \((e\left(\frac{11}{36}\right),e\left(\frac{59}{108}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4033 }(1271, a) \) \(1\)\(1\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{19}{54}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{59}{108}\right)\)\(e\left(\frac{43}{54}\right)\)\(e\left(\frac{17}{27}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{19}{27}\right)\)\(e\left(\frac{107}{108}\right)\)\(e\left(\frac{55}{108}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4033 }(1271,a) \;\) at \(\;a = \) e.g. 2