Properties

Label 4033.870
Modulus $4033$
Conductor $4033$
Order $36$
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(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([35,1]))
 
pari: [g,chi] = znchar(Mod(870,4033))
 

Basic properties

Modulus: \(4033\)
Conductor: \(4033\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
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.ex

\(\chi_{4033}(346,\cdot)\) \(\chi_{4033}(870,\cdot)\) \(\chi_{4033}(1058,\cdot)\) \(\chi_{4033}(1145,\cdot)\) \(\chi_{4033}(1216,\cdot)\) \(\chi_{4033}(1830,\cdot)\) \(\chi_{4033}(2203,\cdot)\) \(\chi_{4033}(2817,\cdot)\) \(\chi_{4033}(2888,\cdot)\) \(\chi_{4033}(2975,\cdot)\) \(\chi_{4033}(3163,\cdot)\) \(\chi_{4033}(3687,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: Number field defined by a degree 36 polynomial

Values on generators

\((1963,2295)\) → \((e\left(\frac{35}{36}\right),e\left(\frac{1}{36}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4033 }(870, a) \) \(1\)\(1\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{17}{36}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{17}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4033 }(870,a) \;\) at \(\;a = \) e.g. 2