Properties

Conductor 573
Order 190
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 573.n

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(573)
 
sage: chi = H[218]
 
pari: [g,chi] = znchar(Mod(218,573))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 573
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 190
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 = odd
Orbit label = 573.n
Orbit index = 14

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 ]
 

\(\chi_{573}(2,\cdot)\) \(\chi_{573}(8,\cdot)\) \(\chi_{573}(17,\cdot)\) \(\chi_{573}(20,\cdot)\) \(\chi_{573}(23,\cdot)\) \(\chi_{573}(26,\cdot)\) \(\chi_{573}(50,\cdot)\) \(\chi_{573}(59,\cdot)\) \(\chi_{573}(65,\cdot)\) \(\chi_{573}(68,\cdot)\) \(\chi_{573}(77,\cdot)\) \(\chi_{573}(80,\cdot)\) \(\chi_{573}(86,\cdot)\) \(\chi_{573}(92,\cdot)\) \(\chi_{573}(98,\cdot)\) \(\chi_{573}(104,\cdot)\) \(\chi_{573}(128,\cdot)\) \(\chi_{573}(134,\cdot)\) \(\chi_{573}(149,\cdot)\) \(\chi_{573}(158,\cdot)\) \(\chi_{573}(170,\cdot)\) \(\chi_{573}(194,\cdot)\) \(\chi_{573}(200,\cdot)\) \(\chi_{573}(203,\cdot)\) \(\chi_{573}(206,\cdot)\) \(\chi_{573}(209,\cdot)\) \(\chi_{573}(215,\cdot)\) \(\chi_{573}(218,\cdot)\) \(\chi_{573}(236,\cdot)\) \(\chi_{573}(239,\cdot)\) ...

Values on generators

\((383,19)\) → \((-1,e\left(\frac{79}{95}\right))\)

Values

-11245781011131416
\(-1\)\(1\)\(e\left(\frac{17}{190}\right)\)\(e\left(\frac{17}{95}\right)\)\(e\left(\frac{3}{38}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{51}{190}\right)\)\(e\left(\frac{16}{95}\right)\)\(e\left(\frac{7}{38}\right)\)\(e\left(\frac{13}{95}\right)\)\(e\left(\frac{11}{38}\right)\)\(e\left(\frac{34}{95}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{95})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 573 }(218,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{573}(218,\cdot)) = \sum_{r\in \Z/573\Z} \chi_{573}(218,r) e\left(\frac{2r}{573}\right) = -23.7774777703+-2.7625261776i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 573 }(218,·),\chi_{ 573 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{573}(218,\cdot),\chi_{573}(1,\cdot)) = \sum_{r\in \Z/573\Z} \chi_{573}(218,r) \chi_{573}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 573 }(218,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{573}(218,·)) = \sum_{r \in \Z/573\Z} \chi_{573}(218,r) e\left(\frac{1 r + 2 r^{-1}}{573}\right) = 0.0 \)