Properties

Label 8002.583
Modulus $8002$
Conductor $4001$
Order $100$
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(100))
 
M = H._module
 
chi = DirichletCharacter(H, M([33]))
 
pari: [g,chi] = znchar(Mod(583,8002))
 

Basic properties

Modulus: \(8002\)
Conductor: \(4001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(100\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4001}(583,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8002.n

\(\chi_{8002}(25,\cdot)\) \(\chi_{8002}(379,\cdot)\) \(\chi_{8002}(579,\cdot)\) \(\chi_{8002}(583,\cdot)\) \(\chi_{8002}(1735,\cdot)\) \(\chi_{8002}(1761,\cdot)\) \(\chi_{8002}(1773,\cdot)\) \(\chi_{8002}(1785,\cdot)\) \(\chi_{8002}(1873,\cdot)\) \(\chi_{8002}(2333,\cdot)\) \(\chi_{8002}(2545,\cdot)\) \(\chi_{8002}(2847,\cdot)\) \(\chi_{8002}(2977,\cdot)\) \(\chi_{8002}(3017,\cdot)\) \(\chi_{8002}(3347,\cdot)\) \(\chi_{8002}(3353,\cdot)\) \(\chi_{8002}(3651,\cdot)\) \(\chi_{8002}(3717,\cdot)\) \(\chi_{8002}(3841,\cdot)\) \(\chi_{8002}(3897,\cdot)\) \(\chi_{8002}(4105,\cdot)\) \(\chi_{8002}(4161,\cdot)\) \(\chi_{8002}(4285,\cdot)\) \(\chi_{8002}(4351,\cdot)\) \(\chi_{8002}(4649,\cdot)\) \(\chi_{8002}(4655,\cdot)\) \(\chi_{8002}(4985,\cdot)\) \(\chi_{8002}(5025,\cdot)\) \(\chi_{8002}(5155,\cdot)\) \(\chi_{8002}(5457,\cdot)\) ...

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: $\Q(\zeta_{100})$
Fixed field: Number field defined by a degree 100 polynomial

Values on generators

\(3\) → \(e\left(\frac{33}{100}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 8002 }(583, a) \) \(1\)\(1\)\(e\left(\frac{33}{100}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{6}{25}\right)\)\(e\left(\frac{33}{50}\right)\)\(i\)\(e\left(\frac{3}{25}\right)\)\(e\left(\frac{73}{100}\right)\)\(e\left(\frac{23}{100}\right)\)\(e\left(\frac{23}{25}\right)\)\(e\left(\frac{57}{100}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8002 }(583,a) \;\) at \(\;a = \) e.g. 2