Properties

Label 864.743
Modulus $864$
Conductor $432$
Order $36$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(864, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([18,9,34]))
 
pari: [g,chi] = znchar(Mod(743,864))
 

Basic properties

Modulus: \(864\)
Conductor: \(432\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{432}(203,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 864.br

\(\chi_{864}(23,\cdot)\) \(\chi_{864}(119,\cdot)\) \(\chi_{864}(167,\cdot)\) \(\chi_{864}(263,\cdot)\) \(\chi_{864}(311,\cdot)\) \(\chi_{864}(407,\cdot)\) \(\chi_{864}(455,\cdot)\) \(\chi_{864}(551,\cdot)\) \(\chi_{864}(599,\cdot)\) \(\chi_{864}(695,\cdot)\) \(\chi_{864}(743,\cdot)\) \(\chi_{864}(839,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: 36.36.5532004127928253705369187176396364210546696053048780432717505515499814912.1

Values on generators

\((703,325,353)\) → \((-1,i,e\left(\frac{17}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 864 }(743, a) \) \(1\)\(1\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{7}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 864 }(743,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 864 }(743,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 864 }(743,·),\chi_{ 864 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 864 }(743,·)) \;\) at \(\; a,b = \) e.g. 1,2