Properties

Label 6042.1681
Modulus $6042$
Conductor $1007$
Order $234$
Real no
Primitive no
Minimal yes
Parity even

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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([0,104,171]))
 
pari: [g,chi] = znchar(Mod(1681,6042))
 

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 \(\chi_{1007}(674,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6042.cn

\(\chi_{6042}(25,\cdot)\) \(\chi_{6042}(43,\cdot)\) \(\chi_{6042}(199,\cdot)\) \(\chi_{6042}(271,\cdot)\) \(\chi_{6042}(481,\cdot)\) \(\chi_{6042}(517,\cdot)\) \(\chi_{6042}(541,\cdot)\) \(\chi_{6042}(643,\cdot)\) \(\chi_{6042}(727,\cdot)\) \(\chi_{6042}(859,\cdot)\) \(\chi_{6042}(997,\cdot)\) \(\chi_{6042}(1069,\cdot)\) \(\chi_{6042}(1183,\cdot)\) \(\chi_{6042}(1195,\cdot)\) \(\chi_{6042}(1225,\cdot)\) \(\chi_{6042}(1279,\cdot)\) \(\chi_{6042}(1297,\cdot)\) \(\chi_{6042}(1309,\cdot)\) \(\chi_{6042}(1315,\cdot)\) \(\chi_{6042}(1543,\cdot)\) \(\chi_{6042}(1681,\cdot)\) \(\chi_{6042}(1753,\cdot)\) \(\chi_{6042}(1999,\cdot)\) \(\chi_{6042}(2023,\cdot)\) \(\chi_{6042}(2107,\cdot)\) \(\chi_{6042}(2137,\cdot)\) \(\chi_{6042}(2251,\cdot)\) \(\chi_{6042}(2341,\cdot)\) \(\chi_{6042}(2449,\cdot)\) \(\chi_{6042}(2455,\cdot)\) ...

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

\((2015,4771,2281)\) → \((1,e\left(\frac{4}{9}\right),e\left(\frac{19}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 6042 }(1681, a) \) \(1\)\(1\)\(e\left(\frac{107}{234}\right)\)\(e\left(\frac{35}{39}\right)\)\(e\left(\frac{28}{39}\right)\)\(e\left(\frac{89}{117}\right)\)\(e\left(\frac{88}{117}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{107}{117}\right)\)\(e\left(\frac{20}{117}\right)\)\(e\left(\frac{61}{78}\right)\)\(e\left(\frac{83}{234}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6042 }(1681,a) \;\) at \(\;a = \) e.g. 2