Properties

Label 4000.359
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([50,25,54]))
 
pari: [g,chi] = znchar(Mod(359,4000))
 

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 \(\chi_{2000}(859,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4000.cx

\(\chi_{4000}(39,\cdot)\) \(\chi_{4000}(119,\cdot)\) \(\chi_{4000}(279,\cdot)\) \(\chi_{4000}(359,\cdot)\) \(\chi_{4000}(439,\cdot)\) \(\chi_{4000}(519,\cdot)\) \(\chi_{4000}(679,\cdot)\) \(\chi_{4000}(759,\cdot)\) \(\chi_{4000}(839,\cdot)\) \(\chi_{4000}(919,\cdot)\) \(\chi_{4000}(1079,\cdot)\) \(\chi_{4000}(1159,\cdot)\) \(\chi_{4000}(1239,\cdot)\) \(\chi_{4000}(1319,\cdot)\) \(\chi_{4000}(1479,\cdot)\) \(\chi_{4000}(1559,\cdot)\) \(\chi_{4000}(1639,\cdot)\) \(\chi_{4000}(1719,\cdot)\) \(\chi_{4000}(1879,\cdot)\) \(\chi_{4000}(1959,\cdot)\) \(\chi_{4000}(2039,\cdot)\) \(\chi_{4000}(2119,\cdot)\) \(\chi_{4000}(2279,\cdot)\) \(\chi_{4000}(2359,\cdot)\) \(\chi_{4000}(2439,\cdot)\) \(\chi_{4000}(2519,\cdot)\) \(\chi_{4000}(2679,\cdot)\) \(\chi_{4000}(2759,\cdot)\) \(\chi_{4000}(2839,\cdot)\) \(\chi_{4000}(2919,\cdot)\) ...

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

\((2751,2501,1377)\) → \((-1,i,e\left(\frac{27}{50}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4000 }(359, a) \) \(-1\)\(1\)\(e\left(\frac{3}{100}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{50}\right)\)\(e\left(\frac{79}{100}\right)\)\(e\left(\frac{81}{100}\right)\)\(e\left(\frac{21}{50}\right)\)\(e\left(\frac{97}{100}\right)\)\(e\left(\frac{93}{100}\right)\)\(e\left(\frac{37}{50}\right)\)\(e\left(\frac{9}{100}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4000 }(359,a) \;\) at \(\;a = \) e.g. 2