Properties

Conductor 201
Order 66
Real no
Primitive no
Minimal yes
Parity even
Orbit label 804.ba

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(804)
 
sage: chi = H[245]
 
pari: [g,chi] = znchar(Mod(245,804))
 

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 = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 804.ba
Orbit index = 27

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 ]
 

\(\chi_{804}(41,\cdot)\) \(\chi_{804}(101,\cdot)\) \(\chi_{804}(113,\cdot)\) \(\chi_{804}(185,\cdot)\) \(\chi_{804}(197,\cdot)\) \(\chi_{804}(221,\cdot)\) \(\chi_{804}(233,\cdot)\) \(\chi_{804}(245,\cdot)\) \(\chi_{804}(281,\cdot)\) \(\chi_{804}(329,\cdot)\) \(\chi_{804}(353,\cdot)\) \(\chi_{804}(413,\cdot)\) \(\chi_{804}(497,\cdot)\) \(\chi_{804}(593,\cdot)\) \(\chi_{804}(605,\cdot)\) \(\chi_{804}(653,\cdot)\) \(\chi_{804}(677,\cdot)\) \(\chi_{804}(701,\cdot)\) \(\chi_{804}(749,\cdot)\) \(\chi_{804}(785,\cdot)\)

Values on generators

\((403,269,337)\) → \((1,-1,e\left(\frac{61}{66}\right))\)

Values

-11571113171923252931
\(1\)\(1\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{17}{66}\right)\)\(e\left(\frac{1}{33}\right)\)\(e\left(\frac{37}{66}\right)\)\(e\left(\frac{43}{66}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{25}{66}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{29}{66}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 804 }(245,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{804}(245,\cdot)) = \sum_{r\in \Z/804\Z} \chi_{804}(245,r) e\left(\frac{r}{402}\right) = -17.8283752595+22.0487876221i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 804 }(245,·),\chi_{ 804 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{804}(245,\cdot),\chi_{804}(1,\cdot)) = \sum_{r\in \Z/804\Z} \chi_{804}(245,r) \chi_{804}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 804 }(245,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{804}(245,·)) = \sum_{r \in \Z/804\Z} \chi_{804}(245,r) e\left(\frac{1 r + 2 r^{-1}}{804}\right) = -0.0 \)