Properties

Label 573.359
Modulus $573$
Conductor $573$
Order $190$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(573, base_ring=CyclotomicField(190))
 
M = H._module
 
chi = DirichletCharacter(H, M([95,39]))
 
pari: [g,chi] = znchar(Mod(359,573))
 

Basic properties

Modulus: \(573\)
Conductor: \(573\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(190\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 573.o

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

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{95})$
Fixed field: Number field defined by a degree 190 polynomial (not computed)

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 573 }(359, a) \) \(1\)\(1\)\(e\left(\frac{101}{190}\right)\)\(e\left(\frac{6}{95}\right)\)\(e\left(\frac{29}{38}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{113}{190}\right)\)\(e\left(\frac{28}{95}\right)\)\(e\left(\frac{18}{19}\right)\)\(e\left(\frac{94}{95}\right)\)\(e\left(\frac{12}{19}\right)\)\(e\left(\frac{12}{95}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 573 }(359,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 573 }(359,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 573 }(359,·),\chi_{ 573 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 573 }(359,·)) \;\) at \(\; a,b = \) e.g. 1,2