Properties

Label 4000.461
Modulus $4000$
Conductor $4000$
Order $200$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4000, base_ring=CyclotomicField(200))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,175,192]))
 
pari: [g,chi] = znchar(Mod(461,4000))
 

Basic properties

Modulus: \(4000\)
Conductor: \(4000\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(200\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4000.dd

\(\chi_{4000}(21,\cdot)\) \(\chi_{4000}(61,\cdot)\) \(\chi_{4000}(141,\cdot)\) \(\chi_{4000}(181,\cdot)\) \(\chi_{4000}(221,\cdot)\) \(\chi_{4000}(261,\cdot)\) \(\chi_{4000}(341,\cdot)\) \(\chi_{4000}(381,\cdot)\) \(\chi_{4000}(421,\cdot)\) \(\chi_{4000}(461,\cdot)\) \(\chi_{4000}(541,\cdot)\) \(\chi_{4000}(581,\cdot)\) \(\chi_{4000}(621,\cdot)\) \(\chi_{4000}(661,\cdot)\) \(\chi_{4000}(741,\cdot)\) \(\chi_{4000}(781,\cdot)\) \(\chi_{4000}(821,\cdot)\) \(\chi_{4000}(861,\cdot)\) \(\chi_{4000}(941,\cdot)\) \(\chi_{4000}(981,\cdot)\) \(\chi_{4000}(1021,\cdot)\) \(\chi_{4000}(1061,\cdot)\) \(\chi_{4000}(1141,\cdot)\) \(\chi_{4000}(1181,\cdot)\) \(\chi_{4000}(1221,\cdot)\) \(\chi_{4000}(1261,\cdot)\) \(\chi_{4000}(1341,\cdot)\) \(\chi_{4000}(1381,\cdot)\) \(\chi_{4000}(1421,\cdot)\) \(\chi_{4000}(1461,\cdot)\) ...

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: $\Q(\zeta_{200})$
Fixed field: Number field defined by a degree 200 polynomial (not computed)

Values on generators

\((2751,2501,1377)\) → \((1,e\left(\frac{7}{8}\right),e\left(\frac{24}{25}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4000 }(461, a) \) \(1\)\(1\)\(e\left(\frac{69}{200}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{69}{100}\right)\)\(e\left(\frac{67}{200}\right)\)\(e\left(\frac{113}{200}\right)\)\(e\left(\frac{29}{50}\right)\)\(e\left(\frac{81}{200}\right)\)\(e\left(\frac{139}{200}\right)\)\(e\left(\frac{1}{100}\right)\)\(e\left(\frac{7}{200}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4000 }(461,a) \;\) at \(\;a = \) e.g. 2