Properties

Label 8002.95
Modulus $8002$
Conductor $4001$
Order $250$
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([69]))
 
pari: [g,chi] = znchar(Mod(95,8002))
 

Basic properties

Modulus: \(8002\)
Conductor: \(4001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(250\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4001}(95,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8002.r

\(\chi_{8002}(65,\cdot)\) \(\chi_{8002}(95,\cdot)\) \(\chi_{8002}(107,\cdot)\) \(\chi_{8002}(211,\cdot)\) \(\chi_{8002}(279,\cdot)\) \(\chi_{8002}(557,\cdot)\) \(\chi_{8002}(737,\cdot)\) \(\chi_{8002}(967,\cdot)\) \(\chi_{8002}(1047,\cdot)\) \(\chi_{8002}(1145,\cdot)\) \(\chi_{8002}(1161,\cdot)\) \(\chi_{8002}(1275,\cdot)\) \(\chi_{8002}(1303,\cdot)\) \(\chi_{8002}(1409,\cdot)\) \(\chi_{8002}(1485,\cdot)\) \(\chi_{8002}(1669,\cdot)\) \(\chi_{8002}(1685,\cdot)\) \(\chi_{8002}(1733,\cdot)\) \(\chi_{8002}(1763,\cdot)\) \(\chi_{8002}(1789,\cdot)\) \(\chi_{8002}(1827,\cdot)\) \(\chi_{8002}(1829,\cdot)\) \(\chi_{8002}(1837,\cdot)\) \(\chi_{8002}(2003,\cdot)\) \(\chi_{8002}(2155,\cdot)\) \(\chi_{8002}(2179,\cdot)\) \(\chi_{8002}(2255,\cdot)\) \(\chi_{8002}(2275,\cdot)\) \(\chi_{8002}(2289,\cdot)\) \(\chi_{8002}(2319,\cdot)\) ...

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 250 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{69}{250}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 8002 }(95, a) \) \(1\)\(1\)\(e\left(\frac{69}{250}\right)\)\(e\left(\frac{17}{25}\right)\)\(e\left(\frac{116}{125}\right)\)\(e\left(\frac{69}{125}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{33}{125}\right)\)\(e\left(\frac{239}{250}\right)\)\(e\left(\frac{239}{250}\right)\)\(e\left(\frac{3}{125}\right)\)\(e\left(\frac{51}{250}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8002 }(95,a) \;\) at \(\;a = \) e.g. 2