Properties

Label 41650.18327
Modulus $41650$
Conductor $25$
Order $20$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(41650, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([1,0,0]))
 
pari: [g,chi] = znchar(Mod(18327,41650))
 

Basic properties

Modulus: \(41650\)
Conductor: \(25\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{25}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 41650.cy

\(\chi_{41650}(1667,\cdot)\) \(\chi_{41650}(3333,\cdot)\) \(\chi_{41650}(9997,\cdot)\) \(\chi_{41650}(11663,\cdot)\) \(\chi_{41650}(18327,\cdot)\) \(\chi_{41650}(28323,\cdot)\) \(\chi_{41650}(34987,\cdot)\) \(\chi_{41650}(36653,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: Number field defined by a degree 20 polynomial

Values on generators

\((18327,2551,2451)\) → \((e\left(\frac{1}{20}\right),1,1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(19\)\(23\)\(27\)\(29\)\(31\)\(33\)
\( \chi_{ 41650 }(18327, a) \) \(-1\)\(1\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 41650 }(18327,a) \;\) at \(\;a = \) e.g. 2