Properties

Label 187.141
Modulus $187$
Conductor $187$
Order $80$
Real no
Primitive yes
Minimal yes
Parity odd

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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([48,25]))
 
pari: [g,chi] = znchar(Mod(141,187))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 187.s

\(\chi_{187}(3,\cdot)\) \(\chi_{187}(5,\cdot)\) \(\chi_{187}(14,\cdot)\) \(\chi_{187}(20,\cdot)\) \(\chi_{187}(27,\cdot)\) \(\chi_{187}(31,\cdot)\) \(\chi_{187}(37,\cdot)\) \(\chi_{187}(48,\cdot)\) \(\chi_{187}(58,\cdot)\) \(\chi_{187}(71,\cdot)\) \(\chi_{187}(75,\cdot)\) \(\chi_{187}(80,\cdot)\) \(\chi_{187}(82,\cdot)\) \(\chi_{187}(91,\cdot)\) \(\chi_{187}(92,\cdot)\) \(\chi_{187}(97,\cdot)\) \(\chi_{187}(108,\cdot)\) \(\chi_{187}(113,\cdot)\) \(\chi_{187}(114,\cdot)\) \(\chi_{187}(124,\cdot)\) \(\chi_{187}(125,\cdot)\) \(\chi_{187}(126,\cdot)\) \(\chi_{187}(130,\cdot)\) \(\chi_{187}(141,\cdot)\) \(\chi_{187}(146,\cdot)\) \(\chi_{187}(147,\cdot)\) \(\chi_{187}(148,\cdot)\) \(\chi_{187}(158,\cdot)\) \(\chi_{187}(159,\cdot)\) \(\chi_{187}(163,\cdot)\) ...

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

\((35,122)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{5}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 187 }(141, a) \) \(-1\)\(1\)\(e\left(\frac{39}{40}\right)\)\(e\left(\frac{9}{80}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{77}{80}\right)\)\(e\left(\frac{7}{80}\right)\)\(e\left(\frac{51}{80}\right)\)\(e\left(\frac{37}{40}\right)\)\(e\left(\frac{9}{40}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{1}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 187 }(141,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 187 }(141,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 187 }(141,·),\chi_{ 187 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 187 }(141,·)) \;\) at \(\; a,b = \) e.g. 1,2