Properties

Label 864.227
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([36,27,52]))
 
pari: [g,chi] = znchar(Mod(227,864))
 

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.bt

\(\chi_{864}(11,\cdot)\) \(\chi_{864}(59,\cdot)\) \(\chi_{864}(83,\cdot)\) \(\chi_{864}(131,\cdot)\) \(\chi_{864}(155,\cdot)\) \(\chi_{864}(203,\cdot)\) \(\chi_{864}(227,\cdot)\) \(\chi_{864}(275,\cdot)\) \(\chi_{864}(299,\cdot)\) \(\chi_{864}(347,\cdot)\) \(\chi_{864}(371,\cdot)\) \(\chi_{864}(419,\cdot)\) \(\chi_{864}(443,\cdot)\) \(\chi_{864}(491,\cdot)\) \(\chi_{864}(515,\cdot)\) \(\chi_{864}(563,\cdot)\) \(\chi_{864}(587,\cdot)\) \(\chi_{864}(635,\cdot)\) \(\chi_{864}(659,\cdot)\) \(\chi_{864}(707,\cdot)\) \(\chi_{864}(731,\cdot)\) \(\chi_{864}(779,\cdot)\) \(\chi_{864}(803,\cdot)\) \(\chi_{864}(851,\cdot)\)

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

\((703,325,353)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{13}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 864 }(227, a) \) \(1\)\(1\)\(e\left(\frac{71}{72}\right)\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{55}{72}\right)\)\(e\left(\frac{29}{72}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{61}{72}\right)\)\(e\left(\frac{17}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 864 }(227,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 864 }(227,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 864 }(227,·),\chi_{ 864 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 864 }(227,·)) \;\) at \(\; a,b = \) e.g. 1,2