Properties

Conductor 201
Order 66
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 201.p

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(201)
 
sage: chi = H[41]
 
pari: [g,chi] = znchar(Mod(41,201))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 201
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 66
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 = 201.p
Orbit index = 16

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 ]
 

\(\chi_{201}(2,\cdot)\) \(\chi_{201}(11,\cdot)\) \(\chi_{201}(20,\cdot)\) \(\chi_{201}(32,\cdot)\) \(\chi_{201}(41,\cdot)\) \(\chi_{201}(44,\cdot)\) \(\chi_{201}(50,\cdot)\) \(\chi_{201}(74,\cdot)\) \(\chi_{201}(80,\cdot)\) \(\chi_{201}(95,\cdot)\) \(\chi_{201}(98,\cdot)\) \(\chi_{201}(101,\cdot)\) \(\chi_{201}(113,\cdot)\) \(\chi_{201}(128,\cdot)\) \(\chi_{201}(146,\cdot)\) \(\chi_{201}(152,\cdot)\) \(\chi_{201}(182,\cdot)\) \(\chi_{201}(185,\cdot)\) \(\chi_{201}(191,\cdot)\) \(\chi_{201}(197,\cdot)\)

Values on generators

\((68,136)\) → \((-1,e\left(\frac{53}{66}\right))\)

Values

-11245781011131416
\(1\)\(1\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{31}{66}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{17}{66}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{7}{33}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{33})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 201 }(41,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{201}(41,\cdot)) = \sum_{r\in \Z/201\Z} \chi_{201}(41,r) e\left(\frac{2r}{201}\right) = 2.8361122205+-13.8908771312i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 201 }(41,·),\chi_{ 201 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{201}(41,\cdot),\chi_{201}(1,\cdot)) = \sum_{r\in \Z/201\Z} \chi_{201}(41,r) \chi_{201}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 201 }(41,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{201}(41,·)) = \sum_{r \in \Z/201\Z} \chi_{201}(41,r) e\left(\frac{1 r + 2 r^{-1}}{201}\right) = 0.0 \)