Properties

Label 799.2
Modulus $799$
Conductor $799$
Order $184$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(799, base_ring=CyclotomicField(184))
 
M = H._module
 
chi = DirichletCharacter(H, M([161,72]))
 
pari: [g,chi] = znchar(Mod(2,799))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 799.r

\(\chi_{799}(2,\cdot)\) \(\chi_{799}(8,\cdot)\) \(\chi_{799}(9,\cdot)\) \(\chi_{799}(25,\cdot)\) \(\chi_{799}(32,\cdot)\) \(\chi_{799}(36,\cdot)\) \(\chi_{799}(42,\cdot)\) \(\chi_{799}(49,\cdot)\) \(\chi_{799}(53,\cdot)\) \(\chi_{799}(59,\cdot)\) \(\chi_{799}(83,\cdot)\) \(\chi_{799}(100,\cdot)\) \(\chi_{799}(110,\cdot)\) \(\chi_{799}(111,\cdot)\) \(\chi_{799}(121,\cdot)\) \(\chi_{799}(128,\cdot)\) \(\chi_{799}(144,\cdot)\) \(\chi_{799}(145,\cdot)\) \(\chi_{799}(155,\cdot)\) \(\chi_{799}(162,\cdot)\) \(\chi_{799}(168,\cdot)\) \(\chi_{799}(178,\cdot)\) \(\chi_{799}(195,\cdot)\) \(\chi_{799}(196,\cdot)\) \(\chi_{799}(202,\cdot)\) \(\chi_{799}(206,\cdot)\) \(\chi_{799}(212,\cdot)\) \(\chi_{799}(213,\cdot)\) \(\chi_{799}(230,\cdot)\) \(\chi_{799}(247,\cdot)\) ...

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

Values on generators

\((377,52)\) → \((e\left(\frac{7}{8}\right),e\left(\frac{9}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 799 }(2, a) \) \(1\)\(1\)\(e\left(\frac{27}{92}\right)\)\(e\left(\frac{129}{184}\right)\)\(e\left(\frac{27}{46}\right)\)\(e\left(\frac{141}{184}\right)\)\(e\left(\frac{183}{184}\right)\)\(e\left(\frac{27}{184}\right)\)\(e\left(\frac{81}{92}\right)\)\(e\left(\frac{37}{92}\right)\)\(e\left(\frac{11}{184}\right)\)\(e\left(\frac{159}{184}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 799 }(2,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

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

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 799 }(2,·),\chi_{ 799 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

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