Properties

Label 8002.2163
Modulus $8002$
Conductor $4001$
Order $125$
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(250))
 
M = H._module
 
chi = DirichletCharacter(H, M([152]))
 
pari: [g,chi] = znchar(Mod(2163,8002))
 

Basic properties

Modulus: \(8002\)
Conductor: \(4001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(125\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4001}(2163,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8002.o

\(\chi_{8002}(35,\cdot)\) \(\chi_{8002}(71,\cdot)\) \(\chi_{8002}(405,\cdot)\) \(\chi_{8002}(417,\cdot)\) \(\chi_{8002}(463,\cdot)\) \(\chi_{8002}(501,\cdot)\) \(\chi_{8002}(615,\cdot)\) \(\chi_{8002}(785,\cdot)\) \(\chi_{8002}(865,\cdot)\) \(\chi_{8002}(1013,\cdot)\) \(\chi_{8002}(1023,\cdot)\) \(\chi_{8002}(1095,\cdot)\) \(\chi_{8002}(1219,\cdot)\) \(\chi_{8002}(1225,\cdot)\) \(\chi_{8002}(1385,\cdot)\) \(\chi_{8002}(1535,\cdot)\) \(\chi_{8002}(1823,\cdot)\) \(\chi_{8002}(1913,\cdot)\) \(\chi_{8002}(2131,\cdot)\) \(\chi_{8002}(2163,\cdot)\) \(\chi_{8002}(2485,\cdot)\) \(\chi_{8002}(2499,\cdot)\) \(\chi_{8002}(2539,\cdot)\) \(\chi_{8002}(2567,\cdot)\) \(\chi_{8002}(2655,\cdot)\) \(\chi_{8002}(2855,\cdot)\) \(\chi_{8002}(2859,\cdot)\) \(\chi_{8002}(2865,\cdot)\) \(\chi_{8002}(2939,\cdot)\) \(\chi_{8002}(3007,\cdot)\) ...

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: $\Q(\zeta_{125})$
Fixed field: Number field defined by a degree 125 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{76}{125}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 8002 }(2163, a) \) \(1\)\(1\)\(e\left(\frac{76}{125}\right)\)\(e\left(\frac{11}{25}\right)\)\(e\left(\frac{78}{125}\right)\)\(e\left(\frac{27}{125}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{89}{125}\right)\)\(e\left(\frac{6}{125}\right)\)\(e\left(\frac{6}{125}\right)\)\(e\left(\frac{99}{125}\right)\)\(e\left(\frac{29}{125}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8002 }(2163,a) \;\) at \(\;a = \) e.g. 2