Properties

Label 8002.253
Modulus $8002$
Conductor $4001$
Order $2000$
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(2000))
 
M = H._module
 
chi = DirichletCharacter(H, M([557]))
 
pari: [g,chi] = znchar(Mod(253,8002))
 

Basic properties

Modulus: \(8002\)
Conductor: \(4001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2000\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4001}(253,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8002.w

\(\chi_{8002}(9,\cdot)\) \(\chi_{8002}(29,\cdot)\) \(\chi_{8002}(31,\cdot)\) \(\chi_{8002}(33,\cdot)\) \(\chi_{8002}(45,\cdot)\) \(\chi_{8002}(47,\cdot)\) \(\chi_{8002}(61,\cdot)\) \(\chi_{8002}(83,\cdot)\) \(\chi_{8002}(89,\cdot)\) \(\chi_{8002}(97,\cdot)\) \(\chi_{8002}(111,\cdot)\) \(\chi_{8002}(117,\cdot)\) \(\chi_{8002}(137,\cdot)\) \(\chi_{8002}(145,\cdot)\) \(\chi_{8002}(149,\cdot)\) \(\chi_{8002}(155,\cdot)\) \(\chi_{8002}(165,\cdot)\) \(\chi_{8002}(171,\cdot)\) \(\chi_{8002}(197,\cdot)\) \(\chi_{8002}(203,\cdot)\) \(\chi_{8002}(225,\cdot)\) \(\chi_{8002}(231,\cdot)\) \(\chi_{8002}(235,\cdot)\) \(\chi_{8002}(253,\cdot)\) \(\chi_{8002}(257,\cdot)\) \(\chi_{8002}(269,\cdot)\) \(\chi_{8002}(289,\cdot)\) \(\chi_{8002}(305,\cdot)\) \(\chi_{8002}(339,\cdot)\) \(\chi_{8002}(377,\cdot)\) ...

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: $\Q(\zeta_{2000})$
Fixed field: Number field defined by a degree 2000 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{557}{2000}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 8002 }(253, a) \) \(1\)\(1\)\(e\left(\frac{557}{2000}\right)\)\(e\left(\frac{63}{100}\right)\)\(e\left(\frac{249}{500}\right)\)\(e\left(\frac{557}{1000}\right)\)\(e\left(\frac{69}{80}\right)\)\(e\left(\frac{337}{500}\right)\)\(e\left(\frac{1817}{2000}\right)\)\(e\left(\frac{1067}{2000}\right)\)\(e\left(\frac{417}{500}\right)\)\(e\left(\frac{1553}{2000}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8002 }(253,a) \;\) at \(\;a = \) e.g. 2