Properties

Label 1700.123
Modulus $1700$
Conductor $1700$
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(1700, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,11,15]))
 
pari: [g,chi] = znchar(Mod(123,1700))
 

Basic properties

Modulus: \(1700\)
Conductor: \(1700\)
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 1700.cd

\(\chi_{1700}(47,\cdot)\) \(\chi_{1700}(123,\cdot)\) \(\chi_{1700}(387,\cdot)\) \(\chi_{1700}(463,\cdot)\) \(\chi_{1700}(727,\cdot)\) \(\chi_{1700}(803,\cdot)\) \(\chi_{1700}(1067,\cdot)\) \(\chi_{1700}(1483,\cdot)\)

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: 20.20.8735421910125170266723632812500000000000000000000.2

Values on generators

\((851,477,1601)\) → \((-1,e\left(\frac{11}{20}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 1700 }(123, a) \) \(1\)\(1\)\(e\left(\frac{1}{10}\right)\)\(-1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{17}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1700 }(123,a) \;\) at \(\;a = \) e.g. 2