Properties

Conductor 573
Order 190
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 573.o

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[359]
 
pari: [g,chi] = znchar(Mod(359,573))
 

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 = even
Orbit label = 573.o
Orbit index = 15

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 ]
 

\(\chi_{573}(29,\cdot)\) \(\chi_{573}(35,\cdot)\) \(\chi_{573}(44,\cdot)\) \(\chi_{573}(47,\cdot)\) \(\chi_{573}(53,\cdot)\) \(\chi_{573}(56,\cdot)\) \(\chi_{573}(62,\cdot)\) \(\chi_{573}(71,\cdot)\) \(\chi_{573}(74,\cdot)\) \(\chi_{573}(83,\cdot)\) \(\chi_{573}(89,\cdot)\) \(\chi_{573}(95,\cdot)\) \(\chi_{573}(101,\cdot)\) \(\chi_{573}(110,\cdot)\) \(\chi_{573}(113,\cdot)\) \(\chi_{573}(116,\cdot)\) \(\chi_{573}(119,\cdot)\) \(\chi_{573}(131,\cdot)\) \(\chi_{573}(137,\cdot)\) \(\chi_{573}(140,\cdot)\) \(\chi_{573}(143,\cdot)\) \(\chi_{573}(146,\cdot)\) \(\chi_{573}(164,\cdot)\) \(\chi_{573}(167,\cdot)\) \(\chi_{573}(173,\cdot)\) \(\chi_{573}(176,\cdot)\) \(\chi_{573}(179,\cdot)\) \(\chi_{573}(182,\cdot)\) \(\chi_{573}(188,\cdot)\) \(\chi_{573}(212,\cdot)\) ...

Values on generators

\((383,19)\) → \((-1,e\left(\frac{39}{190}\right))\)

Values

-11245781011131416
\(1\)\(1\)\(e\left(\frac{101}{190}\right)\)\(e\left(\frac{6}{95}\right)\)\(e\left(\frac{29}{38}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{113}{190}\right)\)\(e\left(\frac{28}{95}\right)\)\(e\left(\frac{18}{19}\right)\)\(e\left(\frac{94}{95}\right)\)\(e\left(\frac{12}{19}\right)\)\(e\left(\frac{12}{95}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 573 }(359,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{573}(359,\cdot)) = \sum_{r\in \Z/573\Z} \chi_{573}(359,r) e\left(\frac{2r}{573}\right) = -17.1353913565+16.7146152532i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 573 }(359,·),\chi_{ 573 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{573}(359,\cdot),\chi_{573}(1,\cdot)) = \sum_{r\in \Z/573\Z} \chi_{573}(359,r) \chi_{573}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 573 }(359,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{573}(359,·)) = \sum_{r \in \Z/573\Z} \chi_{573}(359,r) e\left(\frac{1 r + 2 r^{-1}}{573}\right) = -0.0 \)