Properties

Label 6042.505
Modulus $6042$
Conductor $1007$
Order $39$
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(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,52,24]))
 
pari: [g,chi] = znchar(Mod(505,6042))
 

Basic properties

Modulus: \(6042\)
Conductor: \(1007\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(39\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1007}(505,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6042.bs

\(\chi_{6042}(49,\cdot)\) \(\chi_{6042}(121,\cdot)\) \(\chi_{6042}(505,\cdot)\) \(\chi_{6042}(577,\cdot)\) \(\chi_{6042}(619,\cdot)\) \(\chi_{6042}(733,\cdot)\) \(\chi_{6042}(805,\cdot)\) \(\chi_{6042}(1075,\cdot)\) \(\chi_{6042}(1261,\cdot)\) \(\chi_{6042}(1531,\cdot)\) \(\chi_{6042}(1603,\cdot)\) \(\chi_{6042}(1759,\cdot)\) \(\chi_{6042}(2215,\cdot)\) \(\chi_{6042}(2401,\cdot)\) \(\chi_{6042}(2515,\cdot)\) \(\chi_{6042}(2557,\cdot)\) \(\chi_{6042}(3355,\cdot)\) \(\chi_{6042}(3469,\cdot)\) \(\chi_{6042}(4339,\cdot)\) \(\chi_{6042}(5137,\cdot)\) \(\chi_{6042}(5293,\cdot)\) \(\chi_{6042}(5593,\cdot)\) \(\chi_{6042}(5707,\cdot)\) \(\chi_{6042}(5821,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 39 polynomial

Values on generators

\((2015,4771,2281)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{4}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 6042 }(505, a) \) \(1\)\(1\)\(e\left(\frac{5}{39}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{28}{39}\right)\)\(e\left(\frac{29}{39}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{10}{39}\right)\)\(e\left(\frac{19}{39}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{17}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6042 }(505,a) \;\) at \(\;a = \) e.g. 2