Properties

Label 6069.16
Modulus $6069$
Conductor $2023$
Order $102$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6069, base_ring=CyclotomicField(102))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,34,81]))
 
pari: [g,chi] = znchar(Mod(16,6069))
 

Basic properties

Modulus: \(6069\)
Conductor: \(2023\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(102\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2023}(16,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6069.cf

\(\chi_{6069}(16,\cdot)\) \(\chi_{6069}(67,\cdot)\) \(\chi_{6069}(373,\cdot)\) \(\chi_{6069}(424,\cdot)\) \(\chi_{6069}(730,\cdot)\) \(\chi_{6069}(781,\cdot)\) \(\chi_{6069}(1087,\cdot)\) \(\chi_{6069}(1138,\cdot)\) \(\chi_{6069}(1495,\cdot)\) \(\chi_{6069}(1801,\cdot)\) \(\chi_{6069}(1852,\cdot)\) \(\chi_{6069}(2158,\cdot)\) \(\chi_{6069}(2209,\cdot)\) \(\chi_{6069}(2515,\cdot)\) \(\chi_{6069}(2566,\cdot)\) \(\chi_{6069}(2872,\cdot)\) \(\chi_{6069}(2923,\cdot)\) \(\chi_{6069}(3229,\cdot)\) \(\chi_{6069}(3280,\cdot)\) \(\chi_{6069}(3586,\cdot)\) \(\chi_{6069}(3637,\cdot)\) \(\chi_{6069}(3943,\cdot)\) \(\chi_{6069}(3994,\cdot)\) \(\chi_{6069}(4300,\cdot)\) \(\chi_{6069}(4351,\cdot)\) \(\chi_{6069}(4657,\cdot)\) \(\chi_{6069}(4708,\cdot)\) \(\chi_{6069}(5014,\cdot)\) \(\chi_{6069}(5065,\cdot)\) \(\chi_{6069}(5371,\cdot)\) ...

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: $\Q(\zeta_{51})$
Fixed field: Number field defined by a degree 102 polynomial (not computed)

Values on generators

\((2024,4336,3760)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{27}{34}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(19\)\(20\)
\( \chi_{ 6069 }(16, a) \) \(1\)\(1\)\(e\left(\frac{28}{51}\right)\)\(e\left(\frac{5}{51}\right)\)\(e\left(\frac{53}{102}\right)\)\(e\left(\frac{11}{17}\right)\)\(e\left(\frac{7}{102}\right)\)\(e\left(\frac{61}{102}\right)\)\(e\left(\frac{11}{17}\right)\)\(e\left(\frac{10}{51}\right)\)\(e\left(\frac{40}{51}\right)\)\(e\left(\frac{21}{34}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6069 }(16,a) \;\) at \(\;a = \) e.g. 2