Properties

Label 731.185
Modulus $731$
Conductor $731$
Order $168$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, base_ring=CyclotomicField(168))
 
M = H._module
 
chi = DirichletCharacter(H, M([63,128]))
 
pari: [g,chi] = znchar(Mod(185,731))
 

Basic properties

Modulus: \(731\)
Conductor: \(731\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(168\)
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 731.bl

\(\chi_{731}(9,\cdot)\) \(\chi_{731}(15,\cdot)\) \(\chi_{731}(25,\cdot)\) \(\chi_{731}(53,\cdot)\) \(\chi_{731}(60,\cdot)\) \(\chi_{731}(66,\cdot)\) \(\chi_{731}(83,\cdot)\) \(\chi_{731}(100,\cdot)\) \(\chi_{731}(110,\cdot)\) \(\chi_{731}(111,\cdot)\) \(\chi_{731}(117,\cdot)\) \(\chi_{731}(138,\cdot)\) \(\chi_{731}(144,\cdot)\) \(\chi_{731}(185,\cdot)\) \(\chi_{731}(189,\cdot)\) \(\chi_{731}(195,\cdot)\) \(\chi_{731}(196,\cdot)\) \(\chi_{731}(212,\cdot)\) \(\chi_{731}(229,\cdot)\) \(\chi_{731}(230,\cdot)\) \(\chi_{731}(240,\cdot)\) \(\chi_{731}(246,\cdot)\) \(\chi_{731}(253,\cdot)\) \(\chi_{731}(281,\cdot)\) \(\chi_{731}(298,\cdot)\) \(\chi_{731}(314,\cdot)\) \(\chi_{731}(315,\cdot)\) \(\chi_{731}(325,\cdot)\) \(\chi_{731}(332,\cdot)\) \(\chi_{731}(359,\cdot)\) ...

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: $\Q(\zeta_{168})$
Fixed field: Number field defined by a degree 168 polynomial (not computed)

Values on generators

\((173,562)\) → \((e\left(\frac{3}{8}\right),e\left(\frac{16}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 731 }(185, a) \) \(1\)\(1\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{23}{168}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{155}{168}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{13}{28}\right)\)\(e\left(\frac{23}{84}\right)\)\(e\left(\frac{125}{168}\right)\)\(e\left(\frac{27}{56}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 731 }(185,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 731 }(185,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 731 }(185,·),\chi_{ 731 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 731 }(185,·)) \;\) at \(\; a,b = \) e.g. 1,2