Properties

Label 6042.409
Modulus $6042$
Conductor $1007$
Order $234$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6042, base_ring=CyclotomicField(234))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,221,171]))
 
pari: [g,chi] = znchar(Mod(409,6042))
 

Basic properties

Modulus: \(6042\)
Conductor: \(1007\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(234\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1007}(409,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6042.cp

\(\chi_{6042}(91,\cdot)\) \(\chi_{6042}(223,\cdot)\) \(\chi_{6042}(241,\cdot)\) \(\chi_{6042}(325,\cdot)\) \(\chi_{6042}(355,\cdot)\) \(\chi_{6042}(409,\cdot)\) \(\chi_{6042}(433,\cdot)\) \(\chi_{6042}(547,\cdot)\) \(\chi_{6042}(661,\cdot)\) \(\chi_{6042}(679,\cdot)\) \(\chi_{6042}(751,\cdot)\) \(\chi_{6042}(865,\cdot)\) \(\chi_{6042}(877,\cdot)\) \(\chi_{6042}(907,\cdot)\) \(\chi_{6042}(979,\cdot)\) \(\chi_{6042}(991,\cdot)\) \(\chi_{6042}(1117,\cdot)\) \(\chi_{6042}(1153,\cdot)\) \(\chi_{6042}(1363,\cdot)\) \(\chi_{6042}(1435,\cdot)\) \(\chi_{6042}(1495,\cdot)\) \(\chi_{6042}(1705,\cdot)\) \(\chi_{6042}(1789,\cdot)\) \(\chi_{6042}(1819,\cdot)\) \(\chi_{6042}(1915,\cdot)\) \(\chi_{6042}(1933,\cdot)\) \(\chi_{6042}(1951,\cdot)\) \(\chi_{6042}(2131,\cdot)\) \(\chi_{6042}(2149,\cdot)\) \(\chi_{6042}(2179,\cdot)\) ...

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: $\Q(\zeta_{117})$
Fixed field: Number field defined by a degree 234 polynomial (not computed)

Values on generators

\((2015,4771,2281)\) → \((1,e\left(\frac{17}{18}\right),e\left(\frac{19}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 6042 }(409, a) \) \(-1\)\(1\)\(e\left(\frac{107}{234}\right)\)\(e\left(\frac{35}{39}\right)\)\(e\left(\frac{28}{39}\right)\)\(e\left(\frac{61}{234}\right)\)\(e\left(\frac{88}{117}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{107}{117}\right)\)\(e\left(\frac{157}{234}\right)\)\(e\left(\frac{11}{39}\right)\)\(e\left(\frac{83}{234}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6042 }(409,a) \;\) at \(\;a = \) e.g. 2