Properties

Label 6042.3959
Modulus $6042$
Conductor $3021$
Order $78$
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(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([39,26,45]))
 
pari: [g,chi] = znchar(Mod(3959,6042))
 

Basic properties

Modulus: \(6042\)
Conductor: \(3021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3021}(938,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6042.cd

\(\chi_{6042}(11,\cdot)\) \(\chi_{6042}(197,\cdot)\) \(\chi_{6042}(467,\cdot)\) \(\chi_{6042}(539,\cdot)\) \(\chi_{6042}(653,\cdot)\) \(\chi_{6042}(695,\cdot)\) \(\chi_{6042}(767,\cdot)\) \(\chi_{6042}(1151,\cdot)\) \(\chi_{6042}(1223,\cdot)\) \(\chi_{6042}(1493,\cdot)\) \(\chi_{6042}(1607,\cdot)\) \(\chi_{6042}(1721,\cdot)\) \(\chi_{6042}(2021,\cdot)\) \(\chi_{6042}(2177,\cdot)\) \(\chi_{6042}(2975,\cdot)\) \(\chi_{6042}(3845,\cdot)\) \(\chi_{6042}(3959,\cdot)\) \(\chi_{6042}(4757,\cdot)\) \(\chi_{6042}(4799,\cdot)\) \(\chi_{6042}(4913,\cdot)\) \(\chi_{6042}(5099,\cdot)\) \(\chi_{6042}(5555,\cdot)\) \(\chi_{6042}(5711,\cdot)\) \(\chi_{6042}(5783,\cdot)\)

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 78 polynomial

Values on generators

\((2015,4771,2281)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{15}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 6042 }(3959, a) \) \(-1\)\(1\)\(e\left(\frac{37}{39}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{25}{26}\right)\)\(e\left(\frac{20}{39}\right)\)\(e\left(\frac{47}{78}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{35}{39}\right)\)\(e\left(\frac{55}{78}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{1}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6042 }(3959,a) \;\) at \(\;a = \) e.g. 2