Properties

Label 4000.1929
Modulus $4000$
Conductor $2000$
Order $100$
Real no
Primitive no
Minimal no
Parity even

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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([0,75,22]))
 
pari: [g,chi] = znchar(Mod(1929,4000))
 

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 \(\chi_{2000}(429,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4000.cn

\(\chi_{4000}(9,\cdot)\) \(\chi_{4000}(89,\cdot)\) \(\chi_{4000}(169,\cdot)\) \(\chi_{4000}(329,\cdot)\) \(\chi_{4000}(409,\cdot)\) \(\chi_{4000}(489,\cdot)\) \(\chi_{4000}(569,\cdot)\) \(\chi_{4000}(729,\cdot)\) \(\chi_{4000}(809,\cdot)\) \(\chi_{4000}(889,\cdot)\) \(\chi_{4000}(969,\cdot)\) \(\chi_{4000}(1129,\cdot)\) \(\chi_{4000}(1209,\cdot)\) \(\chi_{4000}(1289,\cdot)\) \(\chi_{4000}(1369,\cdot)\) \(\chi_{4000}(1529,\cdot)\) \(\chi_{4000}(1609,\cdot)\) \(\chi_{4000}(1689,\cdot)\) \(\chi_{4000}(1769,\cdot)\) \(\chi_{4000}(1929,\cdot)\) \(\chi_{4000}(2009,\cdot)\) \(\chi_{4000}(2089,\cdot)\) \(\chi_{4000}(2169,\cdot)\) \(\chi_{4000}(2329,\cdot)\) \(\chi_{4000}(2409,\cdot)\) \(\chi_{4000}(2489,\cdot)\) \(\chi_{4000}(2569,\cdot)\) \(\chi_{4000}(2729,\cdot)\) \(\chi_{4000}(2809,\cdot)\) \(\chi_{4000}(2889,\cdot)\) ...

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

\((2751,2501,1377)\) → \((1,-i,e\left(\frac{11}{50}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4000 }(1929, a) \) \(1\)\(1\)\(e\left(\frac{79}{100}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{29}{50}\right)\)\(e\left(\frac{47}{100}\right)\)\(e\left(\frac{83}{100}\right)\)\(e\left(\frac{3}{50}\right)\)\(e\left(\frac{21}{100}\right)\)\(e\left(\frac{99}{100}\right)\)\(e\left(\frac{8}{25}\right)\)\(e\left(\frac{37}{100}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4000 }(1929,a) \;\) at \(\;a = \) e.g. 2