Properties

Label 6027.1117
Modulus $6027$
Conductor $2009$
Order $105$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6027, base_ring=CyclotomicField(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,170,42]))
 
pari: [g,chi] = znchar(Mod(1117,6027))
 

Basic properties

Modulus: \(6027\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(105\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2009}(1117,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6027.ds

\(\chi_{6027}(16,\cdot)\) \(\chi_{6027}(37,\cdot)\) \(\chi_{6027}(100,\cdot)\) \(\chi_{6027}(256,\cdot)\) \(\chi_{6027}(529,\cdot)\) \(\chi_{6027}(592,\cdot)\) \(\chi_{6027}(625,\cdot)\) \(\chi_{6027}(877,\cdot)\) \(\chi_{6027}(898,\cdot)\) \(\chi_{6027}(1117,\cdot)\) \(\chi_{6027}(1369,\cdot)\) \(\chi_{6027}(1453,\cdot)\) \(\chi_{6027}(1486,\cdot)\) \(\chi_{6027}(1738,\cdot)\) \(\chi_{6027}(1759,\cdot)\) \(\chi_{6027}(1822,\cdot)\) \(\chi_{6027}(2230,\cdot)\) \(\chi_{6027}(2251,\cdot)\) \(\chi_{6027}(2314,\cdot)\) \(\chi_{6027}(2347,\cdot)\) \(\chi_{6027}(2599,\cdot)\) \(\chi_{6027}(2620,\cdot)\) \(\chi_{6027}(2683,\cdot)\) \(\chi_{6027}(2839,\cdot)\) \(\chi_{6027}(3091,\cdot)\) \(\chi_{6027}(3112,\cdot)\) \(\chi_{6027}(3175,\cdot)\) \(\chi_{6027}(3208,\cdot)\) \(\chi_{6027}(3481,\cdot)\) \(\chi_{6027}(3544,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 105 polynomial (not computed)

Values on generators

\((4019,493,2794)\) → \((1,e\left(\frac{17}{21}\right),e\left(\frac{1}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 6027 }(1117, a) \) \(1\)\(1\)\(e\left(\frac{26}{105}\right)\)\(e\left(\frac{52}{105}\right)\)\(e\left(\frac{92}{105}\right)\)\(e\left(\frac{26}{35}\right)\)\(e\left(\frac{13}{105}\right)\)\(e\left(\frac{103}{105}\right)\)\(e\left(\frac{32}{35}\right)\)\(e\left(\frac{104}{105}\right)\)\(e\left(\frac{88}{105}\right)\)\(e\left(\frac{2}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6027 }(1117,a) \;\) at \(\;a = \) e.g. 2