Properties

Label 573.194
Modulus $573$
Conductor $573$
Order $190$
Real no
Primitive yes
Minimal yes
Parity odd

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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([95,116]))
 
pari: [g,chi] = znchar(Mod(194,573))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 573.n

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

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

\((383,19)\) → \((-1,e\left(\frac{58}{95}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 573 }(194, a) \) \(-1\)\(1\)\(e\left(\frac{69}{190}\right)\)\(e\left(\frac{69}{95}\right)\)\(e\left(\frac{1}{38}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{17}{190}\right)\)\(e\left(\frac{37}{95}\right)\)\(e\left(\frac{15}{38}\right)\)\(e\left(\frac{36}{95}\right)\)\(e\left(\frac{29}{38}\right)\)\(e\left(\frac{43}{95}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 573 }(194,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 573 }(194,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 573 }(194,·),\chi_{ 573 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 573 }(194,·)) \;\) at \(\; a,b = \) e.g. 1,2