Properties

Label 799.29
Modulus $799$
Conductor $799$
Order $368$
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(368))
 
M = H._module
 
chi = DirichletCharacter(H, M([299,280]))
 
pari: [g,chi] = znchar(Mod(29,799))
 

Basic properties

Modulus: \(799\)
Conductor: \(799\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(368\)
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.s

\(\chi_{799}(5,\cdot)\) \(\chi_{799}(10,\cdot)\) \(\chi_{799}(11,\cdot)\) \(\chi_{799}(20,\cdot)\) \(\chi_{799}(22,\cdot)\) \(\chi_{799}(23,\cdot)\) \(\chi_{799}(29,\cdot)\) \(\chi_{799}(31,\cdot)\) \(\chi_{799}(39,\cdot)\) \(\chi_{799}(40,\cdot)\) \(\chi_{799}(41,\cdot)\) \(\chi_{799}(44,\cdot)\) \(\chi_{799}(45,\cdot)\) \(\chi_{799}(57,\cdot)\) \(\chi_{799}(58,\cdot)\) \(\chi_{799}(62,\cdot)\) \(\chi_{799}(73,\cdot)\) \(\chi_{799}(78,\cdot)\) \(\chi_{799}(80,\cdot)\) \(\chi_{799}(82,\cdot)\) \(\chi_{799}(88,\cdot)\) \(\chi_{799}(90,\cdot)\) \(\chi_{799}(91,\cdot)\) \(\chi_{799}(92,\cdot)\) \(\chi_{799}(99,\cdot)\) \(\chi_{799}(105,\cdot)\) \(\chi_{799}(107,\cdot)\) \(\chi_{799}(109,\cdot)\) \(\chi_{799}(113,\cdot)\) \(\chi_{799}(114,\cdot)\) ...

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: $\Q(\zeta_{368})$
Fixed field: Number field defined by a degree 368 polynomial (not computed)

Values on generators

\((377,52)\) → \((e\left(\frac{13}{16}\right),e\left(\frac{35}{46}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 799 }(29, a) \) \(1\)\(1\)\(e\left(\frac{13}{184}\right)\)\(e\left(\frac{11}{368}\right)\)\(e\left(\frac{13}{92}\right)\)\(e\left(\frac{303}{368}\right)\)\(e\left(\frac{37}{368}\right)\)\(e\left(\frac{105}{368}\right)\)\(e\left(\frac{39}{184}\right)\)\(e\left(\frac{11}{184}\right)\)\(e\left(\frac{329}{368}\right)\)\(e\left(\frac{5}{368}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 799 }(29,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 799 }(29,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 799 }(29,·),\chi_{ 799 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 799 }(29,·)) \;\) at \(\; a,b = \) e.g. 1,2