Properties

Label 2001.758
Modulus $2001$
Conductor $2001$
Order $14$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2001, base_ring=CyclotomicField(14))
 
M = H._module
 
chi = DirichletCharacter(H, M([7,7,1]))
 
pari: [g,chi] = znchar(Mod(758,2001))
 

Basic properties

Modulus: \(2001\)
Conductor: \(2001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(14\)
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 2001.o

\(\chi_{2001}(689,\cdot)\) \(\chi_{2001}(758,\cdot)\) \(\chi_{2001}(1310,\cdot)\) \(\chi_{2001}(1517,\cdot)\) \(\chi_{2001}(1724,\cdot)\) \(\chi_{2001}(1862,\cdot)\)

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: \(\Q(\zeta_{7})\)
Fixed field: Number field defined by a degree 14 polynomial

Values on generators

\((668,1132,553)\) → \((-1,-1,e\left(\frac{1}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 2001 }(758, a) \) \(1\)\(1\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{2}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2001 }(758,a) \;\) at \(\;a = \) e.g. 2