Properties

Label 6045.178
Modulus $6045$
Conductor $2015$
Order $60$
Real no
Primitive no
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([0,45,40,54]))
 
pari: [g,chi] = znchar(Mod(178,6045))
 

Basic properties

Modulus: \(6045\)
Conductor: \(2015\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2015}(178,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6045.ms

\(\chi_{6045}(178,\cdot)\) \(\chi_{6045}(523,\cdot)\) \(\chi_{6045}(802,\cdot)\) \(\chi_{6045}(1108,\cdot)\) \(\chi_{6045}(1348,\cdot)\) \(\chi_{6045}(1387,\cdot)\) \(\chi_{6045}(1732,\cdot)\) \(\chi_{6045}(2278,\cdot)\) \(\chi_{6045}(2317,\cdot)\) \(\chi_{6045}(2323,\cdot)\) \(\chi_{6045}(2557,\cdot)\) \(\chi_{6045}(3253,\cdot)\) \(\chi_{6045}(3487,\cdot)\) \(\chi_{6045}(3532,\cdot)\) \(\chi_{6045}(4462,\cdot)\) \(\chi_{6045}(5638,\cdot)\)

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

\((4031,4837,1861,2731)\) → \((1,-i,e\left(\frac{2}{3}\right),e\left(\frac{9}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(14\)\(16\)\(17\)\(19\)\(22\)
\( \chi_{ 6045 }(178, a) \) \(1\)\(1\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{23}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6045 }(178,a) \;\) at \(\;a = \) e.g. 2