Properties

Label 772.703
Modulus $772$
Conductor $772$
Order $32$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(772, base_ring=CyclotomicField(32))
 
M = H._module
 
chi = DirichletCharacter(H, M([16,25]))
 
pari: [g,chi] = znchar(Mod(703,772))
 

Basic properties

Modulus: \(772\)
Conductor: \(772\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
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 772.t

\(\chi_{772}(23,\cdot)\) \(\chi_{772}(67,\cdot)\) \(\chi_{772}(151,\cdot)\) \(\chi_{772}(179,\cdot)\) \(\chi_{772}(207,\cdot)\) \(\chi_{772}(235,\cdot)\) \(\chi_{772}(319,\cdot)\) \(\chi_{772}(363,\cdot)\) \(\chi_{772}(455,\cdot)\) \(\chi_{772}(507,\cdot)\) \(\chi_{772}(555,\cdot)\) \(\chi_{772}(571,\cdot)\) \(\chi_{772}(587,\cdot)\) \(\chi_{772}(603,\cdot)\) \(\chi_{772}(651,\cdot)\) \(\chi_{772}(703,\cdot)\)

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: \(\Q(\zeta_{32})\)
Fixed field: 32.0.305658473945657028577927750534391762814465590760665601328732812917726751722831872.1

Values on generators

\((387,5)\) → \((-1,e\left(\frac{25}{32}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 772 }(703, a) \) \(-1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{25}{32}\right)\)\(-i\)\(i\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{5}{32}\right)\)\(e\left(\frac{29}{32}\right)\)\(e\left(\frac{7}{32}\right)\)\(e\left(\frac{25}{32}\right)\)\(e\left(\frac{7}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 772 }(703,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 772 }(703,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 772 }(703,·),\chi_{ 772 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 772 }(703,·)) \;\) at \(\; a,b = \) e.g. 1,2