Properties

Label 579.260
Modulus $579$
Conductor $579$
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(579, base_ring=CyclotomicField(32))
 
M = H._module
 
chi = DirichletCharacter(H, M([16,3]))
 
pari: [g,chi] = znchar(Mod(260,579))
 

Basic properties

Modulus: \(579\)
Conductor: \(579\)
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 579.s

\(\chi_{579}(8,\cdot)\) \(\chi_{579}(14,\cdot)\) \(\chi_{579}(23,\cdot)\) \(\chi_{579}(170,\cdot)\) \(\chi_{579}(179,\cdot)\) \(\chi_{579}(185,\cdot)\) \(\chi_{579}(260,\cdot)\) \(\chi_{579}(314,\cdot)\) \(\chi_{579}(317,\cdot)\) \(\chi_{579}(344,\cdot)\) \(\chi_{579}(362,\cdot)\) \(\chi_{579}(410,\cdot)\) \(\chi_{579}(428,\cdot)\) \(\chi_{579}(455,\cdot)\) \(\chi_{579}(458,\cdot)\) \(\chi_{579}(512,\cdot)\)

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: 32.0.3063491324247901158147277924095178749079926649475132560984974419980910475284097.1

Values on generators

\((194,391)\) → \((-1,e\left(\frac{3}{32}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 579 }(260, a) \) \(-1\)\(1\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{19}{32}\right)\)\(-i\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{9}{32}\right)\)\(e\left(\frac{21}{32}\right)\)\(e\left(\frac{7}{32}\right)\)\(e\left(\frac{7}{16}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 579 }(260,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 579 }(260,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 579 }(260,·),\chi_{ 579 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 579 }(260,·)) \;\) at \(\; a,b = \) e.g. 1,2