Properties

Label 799.35
Modulus $799$
Conductor $47$
Order $46$
Real no
Primitive no
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([0,33]))
 
pari: [g,chi] = znchar(Mod(35,799))
 

Basic properties

Modulus: \(799\)
Conductor: \(47\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(46\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{47}(35,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 799.l

\(\chi_{799}(35,\cdot)\) \(\chi_{799}(52,\cdot)\) \(\chi_{799}(69,\cdot)\) \(\chi_{799}(86,\cdot)\) \(\chi_{799}(120,\cdot)\) \(\chi_{799}(137,\cdot)\) \(\chi_{799}(154,\cdot)\) \(\chi_{799}(171,\cdot)\) \(\chi_{799}(273,\cdot)\) \(\chi_{799}(358,\cdot)\) \(\chi_{799}(409,\cdot)\) \(\chi_{799}(443,\cdot)\) \(\chi_{799}(511,\cdot)\) \(\chi_{799}(528,\cdot)\) \(\chi_{799}(562,\cdot)\) \(\chi_{799}(579,\cdot)\) \(\chi_{799}(630,\cdot)\) \(\chi_{799}(681,\cdot)\) \(\chi_{799}(698,\cdot)\) \(\chi_{799}(715,\cdot)\) \(\chi_{799}(749,\cdot)\) \(\chi_{799}(783,\cdot)\)

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: \(\Q(\zeta_{47})\)

Values on generators

\((377,52)\) → \((1,e\left(\frac{33}{46}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{21}{23}\right)\)\(e\left(\frac{8}{23}\right)\)\(e\left(\frac{19}{23}\right)\)\(e\left(\frac{33}{46}\right)\)\(e\left(\frac{6}{23}\right)\)\(e\left(\frac{22}{23}\right)\)\(e\left(\frac{17}{23}\right)\)\(e\left(\frac{16}{23}\right)\)\(e\left(\frac{29}{46}\right)\)\(e\left(\frac{1}{46}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 799 }(35,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 799 }(35,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 799 }(35,·),\chi_{ 799 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 799 }(35,·)) \;\) at \(\; a,b = \) e.g. 1,2