Properties

Label 9200.91
Modulus $9200$
Conductor $9200$
Order $20$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9200, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,5,4,10]))
 
pari: [g,chi] = znchar(Mod(91,9200))
 

Basic properties

Modulus: \(9200\)
Conductor: \(9200\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
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 9200.cm

\(\chi_{9200}(91,\cdot)\) \(\chi_{9200}(1011,\cdot)\) \(\chi_{9200}(1931,\cdot)\) \(\chi_{9200}(3771,\cdot)\) \(\chi_{9200}(4691,\cdot)\) \(\chi_{9200}(5611,\cdot)\) \(\chi_{9200}(6531,\cdot)\) \(\chi_{9200}(8371,\cdot)\)

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

\((1151,6901,2577,1201)\) → \((-1,i,e\left(\frac{1}{5}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\( \chi_{ 9200 }(91, a) \) \(1\)\(1\)\(e\left(\frac{13}{20}\right)\)\(-1\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{3}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9200 }(91,a) \;\) at \(\;a = \) e.g. 2