Properties

Label 847.13
Modulus $847$
Conductor $847$
Order $110$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(847, base_ring=CyclotomicField(110))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([55,101]))
 
pari: [g,chi] = znchar(Mod(13,847))
 

Basic properties

Modulus: \(847\)
Conductor: \(847\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
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 847.ba

\(\chi_{847}(6,\cdot)\) \(\chi_{847}(13,\cdot)\) \(\chi_{847}(41,\cdot)\) \(\chi_{847}(62,\cdot)\) \(\chi_{847}(83,\cdot)\) \(\chi_{847}(90,\cdot)\) \(\chi_{847}(139,\cdot)\) \(\chi_{847}(160,\cdot)\) \(\chi_{847}(167,\cdot)\) \(\chi_{847}(195,\cdot)\) \(\chi_{847}(216,\cdot)\) \(\chi_{847}(237,\cdot)\) \(\chi_{847}(244,\cdot)\) \(\chi_{847}(272,\cdot)\) \(\chi_{847}(293,\cdot)\) \(\chi_{847}(314,\cdot)\) \(\chi_{847}(321,\cdot)\) \(\chi_{847}(349,\cdot)\) \(\chi_{847}(370,\cdot)\) \(\chi_{847}(391,\cdot)\) \(\chi_{847}(398,\cdot)\) \(\chi_{847}(426,\cdot)\) \(\chi_{847}(447,\cdot)\) \(\chi_{847}(468,\cdot)\) \(\chi_{847}(503,\cdot)\) \(\chi_{847}(545,\cdot)\) \(\chi_{847}(552,\cdot)\) \(\chi_{847}(580,\cdot)\) \(\chi_{847}(601,\cdot)\) \(\chi_{847}(622,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

\((122,365)\) → \((-1,e\left(\frac{101}{110}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\(1\)\(1\)\(e\left(\frac{101}{110}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{46}{55}\right)\)\(e\left(\frac{49}{110}\right)\)\(e\left(\frac{12}{55}\right)\)\(e\left(\frac{83}{110}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{13}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 847 }(13,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 847 }(13,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 847 }(13,·),\chi_{ 847 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 847 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2