Properties

Label 6027.3431
Modulus $6027$
Conductor $123$
Order $40$
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(40))
 
M = H._module
 
chi = DirichletCharacter(H, M([20,0,11]))
 
pari: [g,chi] = znchar(Mod(3431,6027))
 

Basic properties

Modulus: \(6027\)
Conductor: \(123\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{123}(110,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6027.co

\(\chi_{6027}(785,\cdot)\) \(\chi_{6027}(932,\cdot)\) \(\chi_{6027}(1079,\cdot)\) \(\chi_{6027}(1961,\cdot)\) \(\chi_{6027}(2108,\cdot)\) \(\chi_{6027}(2402,\cdot)\) \(\chi_{6027}(2549,\cdot)\) \(\chi_{6027}(3431,\cdot)\) \(\chi_{6027}(3578,\cdot)\) \(\chi_{6027}(3725,\cdot)\) \(\chi_{6027}(4607,\cdot)\) \(\chi_{6027}(4901,\cdot)\) \(\chi_{6027}(5195,\cdot)\) \(\chi_{6027}(5342,\cdot)\) \(\chi_{6027}(5636,\cdot)\) \(\chi_{6027}(5930,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: \(\Q(\zeta_{123})^+\)

Values on generators

\((4019,493,2794)\) → \((-1,1,e\left(\frac{11}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 6027 }(3431, a) \) \(1\)\(1\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{13}{40}\right)\)\(e\left(\frac{21}{40}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{23}{40}\right)\)\(e\left(\frac{19}{40}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6027 }(3431,a) \;\) at \(\;a = \) e.g. 2