Properties

Label 8002.6697
Modulus $8002$
Conductor $4001$
Order $20$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8002, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([7]))
 
pari: [g,chi] = znchar(Mod(6697,8002))
 

Basic properties

Modulus: \(8002\)
Conductor: \(4001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4001}(2696,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8002.h

\(\chi_{8002}(1305,\cdot)\) \(\chi_{8002}(2931,\cdot)\) \(\chi_{8002}(3185,\cdot)\) \(\chi_{8002}(3849,\cdot)\) \(\chi_{8002}(4153,\cdot)\) \(\chi_{8002}(4817,\cdot)\) \(\chi_{8002}(5071,\cdot)\) \(\chi_{8002}(6697,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: Number field defined by a degree 20 polynomial

Values on generators

\(3\) → \(e\left(\frac{7}{20}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 8002 }(6697, a) \) \(1\)\(1\)\(e\left(\frac{7}{20}\right)\)\(1\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(-i\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8002 }(6697,a) \;\) at \(\;a = \) e.g. 2