Properties

Label 804.287
Modulus $804$
Conductor $804$
Order $66$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(804, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([33,33,10]))
 
pari: [g,chi] = znchar(Mod(287,804))
 

Basic properties

Modulus: \(804\)
Conductor: \(804\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
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 804.bd

\(\chi_{804}(23,\cdot)\) \(\chi_{804}(35,\cdot)\) \(\chi_{804}(47,\cdot)\) \(\chi_{804}(71,\cdot)\) \(\chi_{804}(83,\cdot)\) \(\chi_{804}(155,\cdot)\) \(\chi_{804}(167,\cdot)\) \(\chi_{804}(227,\cdot)\) \(\chi_{804}(287,\cdot)\) \(\chi_{804}(323,\cdot)\) \(\chi_{804}(371,\cdot)\) \(\chi_{804}(395,\cdot)\) \(\chi_{804}(419,\cdot)\) \(\chi_{804}(467,\cdot)\) \(\chi_{804}(479,\cdot)\) \(\chi_{804}(575,\cdot)\) \(\chi_{804}(659,\cdot)\) \(\chi_{804}(719,\cdot)\) \(\chi_{804}(743,\cdot)\) \(\chi_{804}(791,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((403,269,337)\) → \((-1,-1,e\left(\frac{5}{33}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 804 }(287, a) \) \(1\)\(1\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{13}{66}\right)\)\(e\left(\frac{1}{66}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{41}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 804 }(287,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 804 }(287,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 804 }(287,·),\chi_{ 804 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 804 }(287,·)) \;\) at \(\; a,b = \) e.g. 1,2