Properties

Label 187.73
Modulus $187$
Conductor $187$
Order $80$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(187, base_ring=CyclotomicField(80))
 
M = H._module
 
chi = DirichletCharacter(H, M([56,25]))
 
pari: [g,chi] = znchar(Mod(73,187))
 

Basic properties

Modulus: \(187\)
Conductor: \(187\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(80\)
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 187.t

\(\chi_{187}(6,\cdot)\) \(\chi_{187}(7,\cdot)\) \(\chi_{187}(24,\cdot)\) \(\chi_{187}(28,\cdot)\) \(\chi_{187}(29,\cdot)\) \(\chi_{187}(39,\cdot)\) \(\chi_{187}(40,\cdot)\) \(\chi_{187}(41,\cdot)\) \(\chi_{187}(46,\cdot)\) \(\chi_{187}(57,\cdot)\) \(\chi_{187}(61,\cdot)\) \(\chi_{187}(62,\cdot)\) \(\chi_{187}(63,\cdot)\) \(\chi_{187}(73,\cdot)\) \(\chi_{187}(74,\cdot)\) \(\chi_{187}(79,\cdot)\) \(\chi_{187}(90,\cdot)\) \(\chi_{187}(95,\cdot)\) \(\chi_{187}(96,\cdot)\) \(\chi_{187}(105,\cdot)\) \(\chi_{187}(107,\cdot)\) \(\chi_{187}(112,\cdot)\) \(\chi_{187}(116,\cdot)\) \(\chi_{187}(129,\cdot)\) \(\chi_{187}(139,\cdot)\) \(\chi_{187}(150,\cdot)\) \(\chi_{187}(156,\cdot)\) \(\chi_{187}(160,\cdot)\) \(\chi_{187}(167,\cdot)\) \(\chi_{187}(173,\cdot)\) ...

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: $\Q(\zeta_{80})$
Fixed field: Number field defined by a degree 80 polynomial

Values on generators

\((35,122)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{5}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 187 }(73, a) \) \(1\)\(1\)\(e\left(\frac{3}{40}\right)\)\(e\left(\frac{73}{80}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{29}{80}\right)\)\(e\left(\frac{79}{80}\right)\)\(e\left(\frac{27}{80}\right)\)\(e\left(\frac{9}{40}\right)\)\(e\left(\frac{33}{40}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{1}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 187 }(73,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 187 }(73,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 187 }(73,·),\chi_{ 187 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 187 }(73,·)) \;\) at \(\; a,b = \) e.g. 1,2