Properties

Label 200.77
Modulus $200$
Conductor $200$
Order $20$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(200, base_ring=CyclotomicField(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,10,1]))
 
pari: [g,chi] = znchar(Mod(77,200))
 

Basic properties

Modulus: \(200\)
Conductor: \(200\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
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 200.x

\(\chi_{200}(13,\cdot)\) \(\chi_{200}(37,\cdot)\) \(\chi_{200}(53,\cdot)\) \(\chi_{200}(77,\cdot)\) \(\chi_{200}(117,\cdot)\) \(\chi_{200}(133,\cdot)\) \(\chi_{200}(173,\cdot)\) \(\chi_{200}(197,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: 20.0.3125000000000000000000000000000000.1

Values on generators

\((151,101,177)\) → \((1,-1,e\left(\frac{1}{20}\right))\)

Values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 200 }(77, a) \) \(-1\)\(1\)\(e\left(\frac{17}{20}\right)\)\(i\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{11}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 200 }(77,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 200 }(77,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 200 }(77,·),\chi_{ 200 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 200 }(77,·)) \;\) at \(\; a,b = \) e.g. 1,2