Properties

Label 5733.176
Modulus $5733$
Conductor $5733$
Order $84$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5733, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([70,36,77]))
 
pari: [g,chi] = znchar(Mod(176,5733))
 

Basic properties

Modulus: \(5733\)
Conductor: \(5733\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
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 5733.mn

\(\chi_{5733}(176,\cdot)\) \(\chi_{5733}(470,\cdot)\) \(\chi_{5733}(596,\cdot)\) \(\chi_{5733}(617,\cdot)\) \(\chi_{5733}(995,\cdot)\) \(\chi_{5733}(1289,\cdot)\) \(\chi_{5733}(1415,\cdot)\) \(\chi_{5733}(1436,\cdot)\) \(\chi_{5733}(2234,\cdot)\) \(\chi_{5733}(2633,\cdot)\) \(\chi_{5733}(2927,\cdot)\) \(\chi_{5733}(3053,\cdot)\) \(\chi_{5733}(3074,\cdot)\) \(\chi_{5733}(3452,\cdot)\) \(\chi_{5733}(3746,\cdot)\) \(\chi_{5733}(3893,\cdot)\) \(\chi_{5733}(4271,\cdot)\) \(\chi_{5733}(4565,\cdot)\) \(\chi_{5733}(4691,\cdot)\) \(\chi_{5733}(4712,\cdot)\) \(\chi_{5733}(5090,\cdot)\) \(\chi_{5733}(5384,\cdot)\) \(\chi_{5733}(5510,\cdot)\) \(\chi_{5733}(5531,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((2549,1522,5293)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{3}{7}\right),e\left(\frac{11}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 5733 }(176, a) \) \(1\)\(1\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{71}{84}\right)\)\(e\left(\frac{19}{28}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{53}{84}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5733 }(176,a) \;\) at \(\;a = \) e.g. 2