Properties

Label 8002.129
Modulus $8002$
Conductor $4001$
Order $400$
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(400))
 
M = H._module
 
chi = DirichletCharacter(H, M([331]))
 
pari: [g,chi] = znchar(Mod(129,8002))
 

Basic properties

Modulus: \(8002\)
Conductor: \(4001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(400\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4001}(129,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8002.s

\(\chi_{8002}(59,\cdot)\) \(\chi_{8002}(63,\cdot)\) \(\chi_{8002}(69,\cdot)\) \(\chi_{8002}(127,\cdot)\) \(\chi_{8002}(129,\cdot)\) \(\chi_{8002}(217,\cdot)\) \(\chi_{8002}(223,\cdot)\) \(\chi_{8002}(241,\cdot)\) \(\chi_{8002}(295,\cdot)\) \(\chi_{8002}(303,\cdot)\) \(\chi_{8002}(315,\cdot)\) \(\chi_{8002}(329,\cdot)\) \(\chi_{8002}(345,\cdot)\) \(\chi_{8002}(397,\cdot)\) \(\chi_{8002}(401,\cdot)\) \(\chi_{8002}(409,\cdot)\) \(\chi_{8002}(427,\cdot)\) \(\chi_{8002}(429,\cdot)\) \(\chi_{8002}(581,\cdot)\) \(\chi_{8002}(605,\cdot)\) \(\chi_{8002}(607,\cdot)\) \(\chi_{8002}(623,\cdot)\) \(\chi_{8002}(645,\cdot)\) \(\chi_{8002}(701,\cdot)\) \(\chi_{8002}(829,\cdot)\) \(\chi_{8002}(879,\cdot)\) \(\chi_{8002}(937,\cdot)\) \(\chi_{8002}(963,\cdot)\) \(\chi_{8002}(1085,\cdot)\) \(\chi_{8002}(1139,\cdot)\) ...

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: $\Q(\zeta_{400})$
Fixed field: Number field defined by a degree 400 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{331}{400}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 8002 }(129, a) \) \(1\)\(1\)\(e\left(\frac{331}{400}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{67}{100}\right)\)\(e\left(\frac{131}{200}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{71}{100}\right)\)\(e\left(\frac{111}{400}\right)\)\(e\left(\frac{61}{400}\right)\)\(e\left(\frac{11}{100}\right)\)\(e\left(\frac{199}{400}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8002 }(129,a) \;\) at \(\;a = \) e.g. 2