Properties

Conductor 945
Order 36
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 945.dh

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(945)
 
sage: chi = H[527]
 
pari: [g,chi] = znchar(Mod(527,945))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 945
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 36
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 945.dh
Orbit index = 86

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 ]
 

\(\chi_{945}(23,\cdot)\) \(\chi_{945}(137,\cdot)\) \(\chi_{945}(212,\cdot)\) \(\chi_{945}(263,\cdot)\) \(\chi_{945}(338,\cdot)\) \(\chi_{945}(452,\cdot)\) \(\chi_{945}(527,\cdot)\) \(\chi_{945}(578,\cdot)\) \(\chi_{945}(653,\cdot)\) \(\chi_{945}(767,\cdot)\) \(\chi_{945}(842,\cdot)\) \(\chi_{945}(893,\cdot)\)

Values on generators

\((596,757,136)\) → \((e\left(\frac{17}{18}\right),i,e\left(\frac{1}{3}\right))\)

Values

-1124811131617192223
\(1\)\(1\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{4}{9}\right)\)\(-i\)\(-1\)\(e\left(\frac{17}{36}\right)\)\(e\left(\frac{29}{36}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{36})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 945 }(527,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{945}(527,\cdot)) = \sum_{r\in \Z/945\Z} \chi_{945}(527,r) e\left(\frac{2r}{945}\right) = -30.3886087936+-4.6403077043i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 945 }(527,·),\chi_{ 945 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{945}(527,\cdot),\chi_{945}(1,\cdot)) = \sum_{r\in \Z/945\Z} \chi_{945}(527,r) \chi_{945}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 945 }(527,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{945}(527,·)) = \sum_{r \in \Z/945\Z} \chi_{945}(527,r) e\left(\frac{1 r + 2 r^{-1}}{945}\right) = 0.0 \)