Properties

Conductor 187
Order 40
Real No
Primitive Yes
Parity Even
Orbit Label 187.r

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[9]
 
pari: [g,chi] = znchar(Mod(9,187))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 187
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 40
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.r
Orbit index = 18

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 ]
 

\(\chi_{187}(9,\cdot)\) \(\chi_{187}(15,\cdot)\) \(\chi_{187}(25,\cdot)\) \(\chi_{187}(26,\cdot)\) \(\chi_{187}(36,\cdot)\) \(\chi_{187}(42,\cdot)\) \(\chi_{187}(49,\cdot)\) \(\chi_{187}(53,\cdot)\) \(\chi_{187}(59,\cdot)\) \(\chi_{187}(60,\cdot)\) \(\chi_{187}(70,\cdot)\) \(\chi_{187}(93,\cdot)\) \(\chi_{187}(104,\cdot)\) \(\chi_{187}(168,\cdot)\) \(\chi_{187}(179,\cdot)\) \(\chi_{187}(185,\cdot)\)

Values on generators

\((35,122)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{1}{8}\right))\)

Values

-11234567891012
\(1\)\(1\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{37}{40}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{1}{40}\right)\)\(e\left(\frac{11}{40}\right)\)\(e\left(\frac{23}{40}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{5}{8}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{40})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 187 }(9,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{187}(9,\cdot)) = \sum_{r\in \Z/187\Z} \chi_{187}(9,r) e\left(\frac{2r}{187}\right) = -6.9840683484+-11.7568188429i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 187 }(9,·),\chi_{ 187 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{187}(9,\cdot),\chi_{187}(1,\cdot)) = \sum_{r\in \Z/187\Z} \chi_{187}(9,r) \chi_{187}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 187 }(9,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{187}(9,·)) = \sum_{r \in \Z/187\Z} \chi_{187}(9,r) e\left(\frac{1 r + 2 r^{-1}}{187}\right) = -0.6086816898+-1.1946050788i \)