Properties

Label 9600.1183
Modulus $9600$
Conductor $400$
Order $20$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9600, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,5,0,3]))
 
pari: [g,chi] = znchar(Mod(1183,9600))
 

Basic properties

Modulus: \(9600\)
Conductor: \(400\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{400}(283,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9600.dv

\(\chi_{9600}(1183,\cdot)\) \(\chi_{9600}(2527,\cdot)\) \(\chi_{9600}(3103,\cdot)\) \(\chi_{9600}(4447,\cdot)\) \(\chi_{9600}(5023,\cdot)\) \(\chi_{9600}(6367,\cdot)\) \(\chi_{9600}(8287,\cdot)\) \(\chi_{9600}(8863,\cdot)\)

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: 20.20.104857600000000000000000000000000000000000.1

Values on generators

\((4351,901,6401,5377)\) → \((-1,i,1,e\left(\frac{3}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 9600 }(1183, a) \) \(1\)\(1\)\(-i\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9600 }(1183,a) \;\) at \(\;a = \) e.g. 2