Properties

Label 5733.25
Modulus $5733$
Conductor $5733$
Order $42$
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(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([28,16,21]))
 
pari: [g,chi] = znchar(Mod(25,5733))
 

Basic properties

Modulus: \(5733\)
Conductor: \(5733\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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.km

\(\chi_{5733}(25,\cdot)\) \(\chi_{5733}(688,\cdot)\) \(\chi_{5733}(844,\cdot)\) \(\chi_{5733}(1507,\cdot)\) \(\chi_{5733}(1663,\cdot)\) \(\chi_{5733}(2326,\cdot)\) \(\chi_{5733}(2482,\cdot)\) \(\chi_{5733}(3145,\cdot)\) \(\chi_{5733}(3964,\cdot)\) \(\chi_{5733}(4120,\cdot)\) \(\chi_{5733}(4939,\cdot)\) \(\chi_{5733}(5602,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((2549,1522,5293)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{8}{21}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 5733 }(25, a) \) \(1\)\(1\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5733 }(25,a) \;\) at \(\;a = \) e.g. 2