Properties

Label 5733.202
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([28,54,77]))
 
pari: [g,chi] = znchar(Mod(202,5733))
 

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.mt

\(\chi_{5733}(202,\cdot)\) \(\chi_{5733}(223,\cdot)\) \(\chi_{5733}(349,\cdot)\) \(\chi_{5733}(643,\cdot)\) \(\chi_{5733}(1021,\cdot)\) \(\chi_{5733}(1042,\cdot)\) \(\chi_{5733}(1168,\cdot)\) \(\chi_{5733}(1462,\cdot)\) \(\chi_{5733}(1840,\cdot)\) \(\chi_{5733}(1987,\cdot)\) \(\chi_{5733}(2281,\cdot)\) \(\chi_{5733}(2659,\cdot)\) \(\chi_{5733}(2680,\cdot)\) \(\chi_{5733}(2806,\cdot)\) \(\chi_{5733}(3100,\cdot)\) \(\chi_{5733}(3499,\cdot)\) \(\chi_{5733}(4297,\cdot)\) \(\chi_{5733}(4318,\cdot)\) \(\chi_{5733}(4444,\cdot)\) \(\chi_{5733}(4738,\cdot)\) \(\chi_{5733}(5116,\cdot)\) \(\chi_{5733}(5137,\cdot)\) \(\chi_{5733}(5263,\cdot)\) \(\chi_{5733}(5557,\cdot)\)

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

\((2549,1522,5293)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{9}{14}\right),e\left(\frac{11}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 5733 }(202, a) \) \(1\)\(1\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{47}{84}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{13}{28}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{41}{84}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5733 }(202,a) \;\) at \(\;a = \) e.g. 2