Properties

Label 4000.233
Modulus $4000$
Conductor $2000$
Order $100$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4000, base_ring=CyclotomicField(100))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,75,23]))
 
pari: [g,chi] = znchar(Mod(233,4000))
 

Basic properties

Modulus: \(4000\)
Conductor: \(2000\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(100\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2000}(733,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4000.cp

\(\chi_{4000}(73,\cdot)\) \(\chi_{4000}(217,\cdot)\) \(\chi_{4000}(233,\cdot)\) \(\chi_{4000}(377,\cdot)\) \(\chi_{4000}(537,\cdot)\) \(\chi_{4000}(553,\cdot)\) \(\chi_{4000}(697,\cdot)\) \(\chi_{4000}(713,\cdot)\) \(\chi_{4000}(873,\cdot)\) \(\chi_{4000}(1017,\cdot)\) \(\chi_{4000}(1033,\cdot)\) \(\chi_{4000}(1177,\cdot)\) \(\chi_{4000}(1337,\cdot)\) \(\chi_{4000}(1353,\cdot)\) \(\chi_{4000}(1497,\cdot)\) \(\chi_{4000}(1513,\cdot)\) \(\chi_{4000}(1673,\cdot)\) \(\chi_{4000}(1817,\cdot)\) \(\chi_{4000}(1833,\cdot)\) \(\chi_{4000}(1977,\cdot)\) \(\chi_{4000}(2137,\cdot)\) \(\chi_{4000}(2153,\cdot)\) \(\chi_{4000}(2297,\cdot)\) \(\chi_{4000}(2313,\cdot)\) \(\chi_{4000}(2473,\cdot)\) \(\chi_{4000}(2617,\cdot)\) \(\chi_{4000}(2633,\cdot)\) \(\chi_{4000}(2777,\cdot)\) \(\chi_{4000}(2937,\cdot)\) \(\chi_{4000}(2953,\cdot)\) ...

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

\((2751,2501,1377)\) → \((1,-i,e\left(\frac{23}{100}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4000 }(233, a) \) \(-1\)\(1\)\(e\left(\frac{43}{50}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{18}{25}\right)\)\(e\left(\frac{23}{100}\right)\)\(e\left(\frac{11}{50}\right)\)\(e\left(\frac{79}{100}\right)\)\(e\left(\frac{39}{100}\right)\)\(e\left(\frac{91}{100}\right)\)\(e\left(\frac{63}{100}\right)\)\(e\left(\frac{29}{50}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4000 }(233,a) \;\) at \(\;a = \) e.g. 2