Properties

Label 8002.6815
Modulus $8002$
Conductor $4001$
Order $50$
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(50))
 
M = H._module
 
chi = DirichletCharacter(H, M([11]))
 
pari: [g,chi] = znchar(Mod(6815,8002))
 

Basic properties

Modulus: \(8002\)
Conductor: \(4001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(50\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4001}(2814,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8002.l

\(\chi_{8002}(625,\cdot)\) \(\chi_{8002}(1429,\cdot)\) \(\chi_{8002}(1473,\cdot)\) \(\chi_{8002}(1529,\cdot)\) \(\chi_{8002}(3253,\cdot)\) \(\chi_{8002}(3407,\cdot)\) \(\chi_{8002}(3805,\cdot)\) \(\chi_{8002}(4015,\cdot)\) \(\chi_{8002}(4315,\cdot)\) \(\chi_{8002}(4347,\cdot)\) \(\chi_{8002}(4637,\cdot)\) \(\chi_{8002}(5595,\cdot)\) \(\chi_{8002}(6471,\cdot)\) \(\chi_{8002}(6745,\cdot)\) \(\chi_{8002}(6815,\cdot)\) \(\chi_{8002}(7159,\cdot)\) \(\chi_{8002}(7385,\cdot)\) \(\chi_{8002}(7607,\cdot)\) \(\chi_{8002}(7611,\cdot)\) \(\chi_{8002}(7801,\cdot)\)

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: \(\Q(\zeta_{25})\)
Fixed field: Number field defined by a degree 50 polynomial

Values on generators

\(3\) → \(e\left(\frac{11}{50}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 8002 }(6815, a) \) \(1\)\(1\)\(e\left(\frac{11}{50}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{11}{25}\right)\)\(-1\)\(e\left(\frac{2}{25}\right)\)\(e\left(\frac{41}{50}\right)\)\(e\left(\frac{41}{50}\right)\)\(e\left(\frac{7}{25}\right)\)\(e\left(\frac{19}{50}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8002 }(6815,a) \;\) at \(\;a = \) e.g. 2