Properties

Label 864.133
Modulus $864$
Conductor $864$
Order $72$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(864, base_ring=CyclotomicField(72))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,40]))
 
pari: [g,chi] = znchar(Mod(133,864))
 

Basic properties

Modulus: \(864\)
Conductor: \(864\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(72\)
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 864.bu

\(\chi_{864}(13,\cdot)\) \(\chi_{864}(61,\cdot)\) \(\chi_{864}(85,\cdot)\) \(\chi_{864}(133,\cdot)\) \(\chi_{864}(157,\cdot)\) \(\chi_{864}(205,\cdot)\) \(\chi_{864}(229,\cdot)\) \(\chi_{864}(277,\cdot)\) \(\chi_{864}(301,\cdot)\) \(\chi_{864}(349,\cdot)\) \(\chi_{864}(373,\cdot)\) \(\chi_{864}(421,\cdot)\) \(\chi_{864}(445,\cdot)\) \(\chi_{864}(493,\cdot)\) \(\chi_{864}(517,\cdot)\) \(\chi_{864}(565,\cdot)\) \(\chi_{864}(589,\cdot)\) \(\chi_{864}(637,\cdot)\) \(\chi_{864}(661,\cdot)\) \(\chi_{864}(709,\cdot)\) \(\chi_{864}(733,\cdot)\) \(\chi_{864}(781,\cdot)\) \(\chi_{864}(805,\cdot)\) \(\chi_{864}(853,\cdot)\)

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: $\Q(\zeta_{72})$
Fixed field: Number field defined by a degree 72 polynomial

Values on generators

\((703,325,353)\) → \((1,e\left(\frac{1}{8}\right),e\left(\frac{5}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 864 }(133, a) \) \(1\)\(1\)\(e\left(\frac{65}{72}\right)\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{61}{72}\right)\)\(e\left(\frac{23}{72}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{67}{72}\right)\)\(e\left(\frac{1}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 864 }(133,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 864 }(133,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 864 }(133,·),\chi_{ 864 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 864 }(133,·)) \;\) at \(\; a,b = \) e.g. 1,2