Properties

Label 6042.37
Modulus $6042$
Conductor $1007$
Order $26$
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(26))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,13,15]))
 
pari: [g,chi] = znchar(Mod(37,6042))
 

Basic properties

Modulus: \(6042\)
Conductor: \(1007\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(26\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1007}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6042.bl

\(\chi_{6042}(37,\cdot)\) \(\chi_{6042}(835,\cdot)\) \(\chi_{6042}(1177,\cdot)\) \(\chi_{6042}(1633,\cdot)\) \(\chi_{6042}(1861,\cdot)\) \(\chi_{6042}(2317,\cdot)\) \(\chi_{6042}(2659,\cdot)\) \(\chi_{6042}(2773,\cdot)\) \(\chi_{6042}(2887,\cdot)\) \(\chi_{6042}(3343,\cdot)\) \(\chi_{6042}(4141,\cdot)\) \(\chi_{6042}(5965,\cdot)\)

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: \(\Q(\zeta_{13})\)
Fixed field: Number field defined by a degree 26 polynomial

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 6042 }(37, a) \) \(-1\)\(1\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{10}{13}\right)\)\(-1\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{5}{26}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6042 }(37,a) \;\) at \(\;a = \) e.g. 2