Properties

Label 869.32
Modulus $869$
Conductor $869$
Order $78$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(869, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([39,20]))
 
pari: [g,chi] = znchar(Mod(32,869))
 

Basic properties

Modulus: \(869\)
Conductor: \(869\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 869.x

\(\chi_{869}(32,\cdot)\) \(\chi_{869}(76,\cdot)\) \(\chi_{869}(98,\cdot)\) \(\chi_{869}(208,\cdot)\) \(\chi_{869}(230,\cdot)\) \(\chi_{869}(241,\cdot)\) \(\chi_{869}(263,\cdot)\) \(\chi_{869}(318,\cdot)\) \(\chi_{869}(329,\cdot)\) \(\chi_{869}(406,\cdot)\) \(\chi_{869}(439,\cdot)\) \(\chi_{869}(483,\cdot)\) \(\chi_{869}(494,\cdot)\) \(\chi_{869}(505,\cdot)\) \(\chi_{869}(516,\cdot)\) \(\chi_{869}(593,\cdot)\) \(\chi_{869}(604,\cdot)\) \(\chi_{869}(626,\cdot)\) \(\chi_{869}(637,\cdot)\) \(\chi_{869}(648,\cdot)\) \(\chi_{869}(681,\cdot)\) \(\chi_{869}(736,\cdot)\) \(\chi_{869}(747,\cdot)\) \(\chi_{869}(835,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((475,793)\) → \((-1,e\left(\frac{10}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 869 }(32, a) \) \(-1\)\(1\)\(e\left(\frac{41}{78}\right)\)\(e\left(\frac{10}{39}\right)\)\(e\left(\frac{2}{39}\right)\)\(e\left(\frac{35}{39}\right)\)\(e\left(\frac{61}{78}\right)\)\(e\left(\frac{7}{78}\right)\)\(e\left(\frac{15}{26}\right)\)\(e\left(\frac{20}{39}\right)\)\(e\left(\frac{11}{26}\right)\)\(e\left(\frac{4}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 869 }(32,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 869 }(32,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 869 }(32,·),\chi_{ 869 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 869 }(32,·)) \;\) at \(\; a,b = \) e.g. 1,2