Properties

Label 207.5
Modulus $207$
Conductor $207$
Order $66$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(207, base_ring=CyclotomicField(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([55,3]))
 
pari: [g,chi] = znchar(Mod(5,207))
 

Basic properties

Modulus: \(207\)
Conductor: \(207\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
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 207.o

\(\chi_{207}(5,\cdot)\) \(\chi_{207}(11,\cdot)\) \(\chi_{207}(14,\cdot)\) \(\chi_{207}(20,\cdot)\) \(\chi_{207}(38,\cdot)\) \(\chi_{207}(56,\cdot)\) \(\chi_{207}(65,\cdot)\) \(\chi_{207}(74,\cdot)\) \(\chi_{207}(83,\cdot)\) \(\chi_{207}(86,\cdot)\) \(\chi_{207}(113,\cdot)\) \(\chi_{207}(122,\cdot)\) \(\chi_{207}(149,\cdot)\) \(\chi_{207}(155,\cdot)\) \(\chi_{207}(158,\cdot)\) \(\chi_{207}(176,\cdot)\) \(\chi_{207}(182,\cdot)\) \(\chi_{207}(191,\cdot)\) \(\chi_{207}(194,\cdot)\) \(\chi_{207}(203,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((47,28)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1}{22}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 207 }(5, a) \) \(1\)\(1\)\(e\left(\frac{61}{66}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{13}{66}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{23}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 207 }(5,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 207 }(5,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 207 }(5,·),\chi_{ 207 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 207 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2