Properties

Label 799.13
Modulus $799$
Conductor $799$
Order $92$
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(92))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([23,22]))
 
pari: [g,chi] = znchar(Mod(13,799))
 

Basic properties

Modulus: \(799\)
Conductor: \(799\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(92\)
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.p

\(\chi_{799}(13,\cdot)\) \(\chi_{799}(30,\cdot)\) \(\chi_{799}(38,\cdot)\) \(\chi_{799}(123,\cdot)\) \(\chi_{799}(132,\cdot)\) \(\chi_{799}(174,\cdot)\) \(\chi_{799}(208,\cdot)\) \(\chi_{799}(217,\cdot)\) \(\chi_{799}(268,\cdot)\) \(\chi_{799}(276,\cdot)\) \(\chi_{799}(293,\cdot)\) \(\chi_{799}(302,\cdot)\) \(\chi_{799}(327,\cdot)\) \(\chi_{799}(344,\cdot)\) \(\chi_{799}(370,\cdot)\) \(\chi_{799}(387,\cdot)\) \(\chi_{799}(395,\cdot)\) \(\chi_{799}(421,\cdot)\) \(\chi_{799}(438,\cdot)\) \(\chi_{799}(446,\cdot)\) \(\chi_{799}(463,\cdot)\) \(\chi_{799}(480,\cdot)\) \(\chi_{799}(489,\cdot)\) \(\chi_{799}(514,\cdot)\) \(\chi_{799}(540,\cdot)\) \(\chi_{799}(548,\cdot)\) \(\chi_{799}(557,\cdot)\) \(\chi_{799}(574,\cdot)\) \(\chi_{799}(599,\cdot)\) \(\chi_{799}(608,\cdot)\) ...

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: $\Q(\zeta_{92})$
Fixed field: Number field defined by a degree 92 polynomial

Values on generators

\((377,52)\) → \((i,e\left(\frac{11}{46}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{37}{46}\right)\)\(e\left(\frac{3}{92}\right)\)\(e\left(\frac{14}{23}\right)\)\(e\left(\frac{45}{92}\right)\)\(e\left(\frac{77}{92}\right)\)\(e\left(\frac{37}{92}\right)\)\(e\left(\frac{19}{46}\right)\)\(e\left(\frac{3}{46}\right)\)\(e\left(\frac{27}{92}\right)\)\(e\left(\frac{39}{92}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 799 }(13,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 799 }(13,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 799 }(13,·),\chi_{ 799 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 799 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2