Properties

Label 6045.488
Modulus $6045$
Conductor $6045$
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(6045, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,45,55,54]))
 
pari: [g,chi] = znchar(Mod(488,6045))
 

Basic properties

Modulus: \(6045\)
Conductor: \(6045\)
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 6045.ly

\(\chi_{6045}(488,\cdot)\) \(\chi_{6045}(587,\cdot)\) \(\chi_{6045}(1007,\cdot)\) \(\chi_{6045}(1298,\cdot)\) \(\chi_{6045}(1658,\cdot)\) \(\chi_{6045}(1883,\cdot)\) \(\chi_{6045}(1982,\cdot)\) \(\chi_{6045}(2633,\cdot)\) \(\chi_{6045}(3053,\cdot)\) \(\chi_{6045}(3902,\cdot)\) \(\chi_{6045}(4028,\cdot)\) \(\chi_{6045}(4487,\cdot)\) \(\chi_{6045}(5297,\cdot)\) \(\chi_{6045}(5657,\cdot)\) \(\chi_{6045}(5882,\cdot)\) \(\chi_{6045}(5948,\cdot)\)

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

\((4031,4837,1861,2731)\) → \((-1,-i,e\left(\frac{11}{12}\right),e\left(\frac{9}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(14\)\(16\)\(17\)\(19\)\(22\)
\( \chi_{ 6045 }(488, a) \) \(1\)\(1\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{23}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6045 }(488,a) \;\) at \(\;a = \) e.g. 2