Properties

Label 799.33
Modulus $799$
Conductor $799$
Order $46$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(799, base_ring=CyclotomicField(46))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([23,27]))
 
pari: [g,chi] = znchar(Mod(33,799))
 

Basic properties

Modulus: \(799\)
Conductor: \(799\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(46\)
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 799.m

\(\chi_{799}(33,\cdot)\) \(\chi_{799}(67,\cdot)\) \(\chi_{799}(135,\cdot)\) \(\chi_{799}(152,\cdot)\) \(\chi_{799}(186,\cdot)\) \(\chi_{799}(203,\cdot)\) \(\chi_{799}(254,\cdot)\) \(\chi_{799}(305,\cdot)\) \(\chi_{799}(322,\cdot)\) \(\chi_{799}(339,\cdot)\) \(\chi_{799}(373,\cdot)\) \(\chi_{799}(407,\cdot)\) \(\chi_{799}(458,\cdot)\) \(\chi_{799}(475,\cdot)\) \(\chi_{799}(492,\cdot)\) \(\chi_{799}(509,\cdot)\) \(\chi_{799}(543,\cdot)\) \(\chi_{799}(560,\cdot)\) \(\chi_{799}(577,\cdot)\) \(\chi_{799}(594,\cdot)\) \(\chi_{799}(696,\cdot)\) \(\chi_{799}(781,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{23})\)
Fixed field: 46.0.35053291047098822445710921889723167962319821671472348329989416363445958210915442479511871796900053441791.1

Values on generators

\((377,52)\) → \((-1,e\left(\frac{27}{46}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{13}{23}\right)\)\(e\left(\frac{11}{46}\right)\)\(e\left(\frac{3}{23}\right)\)\(e\left(\frac{2}{23}\right)\)\(e\left(\frac{37}{46}\right)\)\(e\left(\frac{13}{46}\right)\)\(e\left(\frac{16}{23}\right)\)\(e\left(\frac{11}{23}\right)\)\(e\left(\frac{15}{23}\right)\)\(e\left(\frac{14}{23}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 799 }(33,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 799 }(33,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 799 }(33,·),\chi_{ 799 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 799 }(33,·)) \;\) at \(\; a,b = \) e.g. 1,2