Properties

Label 6045.2569
Modulus $6045$
Conductor $2015$
Order $20$
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(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,10,5,2]))
 
pari: [g,chi] = znchar(Mod(2569,6045))
 

Basic properties

Modulus: \(6045\)
Conductor: \(2015\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2015}(554,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6045.ix

\(\chi_{6045}(2569,\cdot)\) \(\chi_{6045}(3034,\cdot)\) \(\chi_{6045}(3154,\cdot)\) \(\chi_{6045}(3619,\cdot)\) \(\chi_{6045}(4324,\cdot)\) \(\chi_{6045}(4789,\cdot)\) \(\chi_{6045}(5299,\cdot)\) \(\chi_{6045}(5764,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: Number field defined by a degree 20 polynomial

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(14\)\(16\)\(17\)\(19\)\(22\)
\( \chi_{ 6045 }(2569, a) \) \(1\)\(1\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{1}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6045 }(2569,a) \;\) at \(\;a = \) e.g. 2