Properties

Label 8020.133
Modulus $8020$
Conductor $2005$
Order $16$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8020, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,12,9]))
 
pari: [g,chi] = znchar(Mod(133,8020))
 

Basic properties

Modulus: \(8020\)
Conductor: \(2005\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2005}(133,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8020.bp

\(\chi_{8020}(133,\cdot)\) \(\chi_{8020}(2273,\cdot)\) \(\chi_{8020}(2553,\cdot)\) \(\chi_{8020}(2777,\cdot)\) \(\chi_{8020}(2837,\cdot)\) \(\chi_{8020}(6217,\cdot)\) \(\chi_{8020}(7417,\cdot)\) \(\chi_{8020}(7873,\cdot)\)

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: \(\Q(\zeta_{16})\)
Fixed field: Number field defined by a degree 16 polynomial

Values on generators

\((4011,6417,7221)\) → \((1,-i,e\left(\frac{9}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 8020 }(133, a) \) \(1\)\(1\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{7}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8020 }(133,a) \;\) at \(\;a = \) e.g. 2