Properties

Label 799.15
Modulus $799$
Conductor $799$
Order $184$
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(184))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([69,84]))
 
pari: [g,chi] = znchar(Mod(15,799))
 

Basic properties

Modulus: \(799\)
Conductor: \(799\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(184\)
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.q

\(\chi_{799}(15,\cdot)\) \(\chi_{799}(19,\cdot)\) \(\chi_{799}(26,\cdot)\) \(\chi_{799}(43,\cdot)\) \(\chi_{799}(60,\cdot)\) \(\chi_{799}(66,\cdot)\) \(\chi_{799}(70,\cdot)\) \(\chi_{799}(76,\cdot)\) \(\chi_{799}(77,\cdot)\) \(\chi_{799}(87,\cdot)\) \(\chi_{799}(104,\cdot)\) \(\chi_{799}(117,\cdot)\) \(\chi_{799}(127,\cdot)\) \(\chi_{799}(134,\cdot)\) \(\chi_{799}(138,\cdot)\) \(\chi_{799}(151,\cdot)\) \(\chi_{799}(161,\cdot)\) \(\chi_{799}(172,\cdot)\) \(\chi_{799}(179,\cdot)\) \(\chi_{799}(185,\cdot)\) \(\chi_{799}(219,\cdot)\) \(\chi_{799}(223,\cdot)\) \(\chi_{799}(229,\cdot)\) \(\chi_{799}(240,\cdot)\) \(\chi_{799}(246,\cdot)\) \(\chi_{799}(257,\cdot)\) \(\chi_{799}(264,\cdot)\) \(\chi_{799}(270,\cdot)\) \(\chi_{799}(274,\cdot)\) \(\chi_{799}(280,\cdot)\) ...

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: $\Q(\zeta_{184})$
Fixed field: Number field defined by a degree 184 polynomial (not computed)

Values on generators

\((377,52)\) → \((e\left(\frac{3}{8}\right),e\left(\frac{21}{46}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{43}{92}\right)\)\(e\left(\frac{93}{184}\right)\)\(e\left(\frac{43}{46}\right)\)\(e\left(\frac{61}{184}\right)\)\(e\left(\frac{179}{184}\right)\)\(e\left(\frac{135}{184}\right)\)\(e\left(\frac{37}{92}\right)\)\(e\left(\frac{1}{92}\right)\)\(e\left(\frac{147}{184}\right)\)\(e\left(\frac{151}{184}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 799 }(15,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 799 }(15,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 799 }(15,·),\chi_{ 799 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 799 }(15,·)) \;\) at \(\; a,b = \) e.g. 1,2