Properties

Label 1600.1247
Modulus $1600$
Conductor $200$
Order $20$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,10,17]))
 
pari: [g,chi] = znchar(Mod(1247,1600))
 

Basic properties

Modulus: \(1600\)
Conductor: \(200\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{200}(147,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1600.bx

\(\chi_{1600}(223,\cdot)\) \(\chi_{1600}(287,\cdot)\) \(\chi_{1600}(863,\cdot)\) \(\chi_{1600}(927,\cdot)\) \(\chi_{1600}(1183,\cdot)\) \(\chi_{1600}(1247,\cdot)\) \(\chi_{1600}(1503,\cdot)\) \(\chi_{1600}(1567,\cdot)\)

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: 20.20.3125000000000000000000000000000000.1

Values on generators

\((1151,901,577)\) → \((-1,-1,e\left(\frac{17}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 1600 }(1247, a) \) \(1\)\(1\)\(e\left(\frac{19}{20}\right)\)\(-i\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{17}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1600 }(1247,a) \;\) at \(\;a = \) e.g. 2