Properties

Conductor 187
Order 80
Real No
Primitive Yes
Parity Even
Orbit Label 187.t

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(187)
 
sage: chi = H[112]
 
pari: [g,chi] = znchar(Mod(112,187))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 187
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 80
Real = No
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = Yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Even
Orbit label = 187.t
Orbit index = 20

Galois orbit

sage: chi.sage_character().galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

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

Values on generators

\((35,122)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{3}{16}\right))\)

Values

-11234567891012
\(1\)\(1\)\(e\left(\frac{29}{40}\right)\)\(e\left(\frac{79}{80}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{27}{80}\right)\)\(e\left(\frac{57}{80}\right)\)\(e\left(\frac{61}{80}\right)\)\(e\left(\frac{7}{40}\right)\)\(e\left(\frac{39}{40}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{7}{16}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{80})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 187 }(112,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{187}(112,\cdot)) = \sum_{r\in \Z/187\Z} \chi_{187}(112,r) e\left(\frac{2r}{187}\right) = 12.7698667243+4.8918814218i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 187 }(112,·),\chi_{ 187 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{187}(112,\cdot),\chi_{187}(1,\cdot)) = \sum_{r\in \Z/187\Z} \chi_{187}(112,r) \chi_{187}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 187 }(112,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{187}(112,·)) = \sum_{r \in \Z/187\Z} \chi_{187}(112,r) e\left(\frac{1 r + 2 r^{-1}}{187}\right) = 0.482037226+-0.5643930769i \)