Properties

Label 4592.187
Modulus $4592$
Conductor $4592$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4592, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,15,50,54]))
 
pari: [g,chi] = znchar(Mod(187,4592))
 

Basic properties

Modulus: \(4592\)
Conductor: \(4592\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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 4592.hn

\(\chi_{4592}(187,\cdot)\) \(\chi_{4592}(523,\cdot)\) \(\chi_{4592}(619,\cdot)\) \(\chi_{4592}(843,\cdot)\) \(\chi_{4592}(1179,\cdot)\) \(\chi_{4592}(1419,\cdot)\) \(\chi_{4592}(2075,\cdot)\) \(\chi_{4592}(2259,\cdot)\) \(\chi_{4592}(2483,\cdot)\) \(\chi_{4592}(2819,\cdot)\) \(\chi_{4592}(2915,\cdot)\) \(\chi_{4592}(3139,\cdot)\) \(\chi_{4592}(3475,\cdot)\) \(\chi_{4592}(3715,\cdot)\) \(\chi_{4592}(4371,\cdot)\) \(\chi_{4592}(4555,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((575,3445,3937,785)\) → \((-1,i,e\left(\frac{5}{6}\right),e\left(\frac{9}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 4592 }(187, a) \) \(1\)\(1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{13}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4592 }(187,a) \;\) at \(\;a = \) e.g. 2