Properties

Label 2000.1341
Modulus $2000$
Conductor $2000$
Order $100$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2000, base_ring=CyclotomicField(100))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,75,24]))
 
pari: [g,chi] = znchar(Mod(1341,2000))
 

Basic properties

Modulus: \(2000\)
Conductor: \(2000\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(100\)
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 2000.cg

\(\chi_{2000}(21,\cdot)\) \(\chi_{2000}(61,\cdot)\) \(\chi_{2000}(141,\cdot)\) \(\chi_{2000}(181,\cdot)\) \(\chi_{2000}(221,\cdot)\) \(\chi_{2000}(261,\cdot)\) \(\chi_{2000}(341,\cdot)\) \(\chi_{2000}(381,\cdot)\) \(\chi_{2000}(421,\cdot)\) \(\chi_{2000}(461,\cdot)\) \(\chi_{2000}(541,\cdot)\) \(\chi_{2000}(581,\cdot)\) \(\chi_{2000}(621,\cdot)\) \(\chi_{2000}(661,\cdot)\) \(\chi_{2000}(741,\cdot)\) \(\chi_{2000}(781,\cdot)\) \(\chi_{2000}(821,\cdot)\) \(\chi_{2000}(861,\cdot)\) \(\chi_{2000}(941,\cdot)\) \(\chi_{2000}(981,\cdot)\) \(\chi_{2000}(1021,\cdot)\) \(\chi_{2000}(1061,\cdot)\) \(\chi_{2000}(1141,\cdot)\) \(\chi_{2000}(1181,\cdot)\) \(\chi_{2000}(1221,\cdot)\) \(\chi_{2000}(1261,\cdot)\) \(\chi_{2000}(1341,\cdot)\) \(\chi_{2000}(1381,\cdot)\) \(\chi_{2000}(1421,\cdot)\) \(\chi_{2000}(1461,\cdot)\) ...

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

Values on generators

\((751,501,1377)\) → \((1,-i,e\left(\frac{6}{25}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 2000 }(1341, a) \) \(1\)\(1\)\(e\left(\frac{93}{100}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{43}{50}\right)\)\(e\left(\frac{99}{100}\right)\)\(e\left(\frac{61}{100}\right)\)\(e\left(\frac{13}{25}\right)\)\(e\left(\frac{57}{100}\right)\)\(e\left(\frac{83}{100}\right)\)\(e\left(\frac{47}{50}\right)\)\(e\left(\frac{79}{100}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2000 }(1341,a) \;\) at \(\;a = \) e.g. 2