Properties

Label 6042.31
Modulus $6042$
Conductor $1007$
Order $156$
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(156))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,130,99]))
 
pari: [g,chi] = znchar(Mod(31,6042))
 

Basic properties

Modulus: \(6042\)
Conductor: \(1007\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(156\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1007}(31,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6042.ch

\(\chi_{6042}(31,\cdot)\) \(\chi_{6042}(103,\cdot)\) \(\chi_{6042}(145,\cdot)\) \(\chi_{6042}(217,\cdot)\) \(\chi_{6042}(373,\cdot)\) \(\chi_{6042}(445,\cdot)\) \(\chi_{6042}(601,\cdot)\) \(\chi_{6042}(715,\cdot)\) \(\chi_{6042}(787,\cdot)\) \(\chi_{6042}(829,\cdot)\) \(\chi_{6042}(1015,\cdot)\) \(\chi_{6042}(1057,\cdot)\) \(\chi_{6042}(1171,\cdot)\) \(\chi_{6042}(1357,\cdot)\) \(\chi_{6042}(1399,\cdot)\) \(\chi_{6042}(1585,\cdot)\) \(\chi_{6042}(1699,\cdot)\) \(\chi_{6042}(1741,\cdot)\) \(\chi_{6042}(1927,\cdot)\) \(\chi_{6042}(1969,\cdot)\) \(\chi_{6042}(2041,\cdot)\) \(\chi_{6042}(2155,\cdot)\) \(\chi_{6042}(2311,\cdot)\) \(\chi_{6042}(2383,\cdot)\) \(\chi_{6042}(2539,\cdot)\) \(\chi_{6042}(2611,\cdot)\) \(\chi_{6042}(2653,\cdot)\) \(\chi_{6042}(2725,\cdot)\) \(\chi_{6042}(2881,\cdot)\) \(\chi_{6042}(2995,\cdot)\) ...

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: $\Q(\zeta_{156})$
Fixed field: Number field defined by a degree 156 polynomial (not computed)

Values on generators

\((2015,4771,2281)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{33}{52}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 6042 }(31, a) \) \(1\)\(1\)\(e\left(\frac{25}{156}\right)\)\(e\left(\frac{23}{26}\right)\)\(e\left(\frac{21}{26}\right)\)\(e\left(\frac{31}{78}\right)\)\(e\left(\frac{53}{78}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{25}{78}\right)\)\(e\left(\frac{14}{39}\right)\)\(e\left(\frac{23}{52}\right)\)\(e\left(\frac{7}{156}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6042 }(31,a) \;\) at \(\;a = \) e.g. 2