Properties

Label 485.369
Modulus $485$
Conductor $485$
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(485, base_ring=CyclotomicField(32))
 
M = H._module
 
chi = DirichletCharacter(H, M([16,11]))
 
pari: [g,chi] = znchar(Mod(369,485))
 

Basic properties

Modulus: \(485\)
Conductor: \(485\)
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 485.bi

\(\chi_{485}(19,\cdot)\) \(\chi_{485}(34,\cdot)\) \(\chi_{485}(69,\cdot)\) \(\chi_{485}(139,\cdot)\) \(\chi_{485}(149,\cdot)\) \(\chi_{485}(164,\cdot)\) \(\chi_{485}(174,\cdot)\) \(\chi_{485}(214,\cdot)\) \(\chi_{485}(224,\cdot)\) \(\chi_{485}(239,\cdot)\) \(\chi_{485}(249,\cdot)\) \(\chi_{485}(319,\cdot)\) \(\chi_{485}(354,\cdot)\) \(\chi_{485}(369,\cdot)\) \(\chi_{485}(434,\cdot)\) \(\chi_{485}(439,\cdot)\)

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: 32.0.5935315803327378381589507037815252283484449810801350950039360504150390625.1

Values on generators

\((292,296)\) → \((-1,e\left(\frac{11}{32}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 485 }(369, a) \) \(-1\)\(1\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{3}{8}\right)\)\(-i\)\(e\left(\frac{5}{32}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{3}{32}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 485 }(369,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 485 }(369,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 485 }(369,·),\chi_{ 485 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 485 }(369,·)) \;\) at \(\; a,b = \) e.g. 1,2