Properties

Label 573.536
Modulus $573$
Conductor $573$
Order $38$
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(38))
 
M = H._module
 
chi = DirichletCharacter(H, M([19,22]))
 
pari: [g,chi] = znchar(Mod(536,573))
 

Basic properties

Modulus: \(573\)
Conductor: \(573\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(38\)
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.l

\(\chi_{573}(5,\cdot)\) \(\chi_{573}(32,\cdot)\) \(\chi_{573}(107,\cdot)\) \(\chi_{573}(125,\cdot)\) \(\chi_{573}(197,\cdot)\) \(\chi_{573}(221,\cdot)\) \(\chi_{573}(227,\cdot)\) \(\chi_{573}(260,\cdot)\) \(\chi_{573}(341,\cdot)\) \(\chi_{573}(344,\cdot)\) \(\chi_{573}(368,\cdot)\) \(\chi_{573}(371,\cdot)\) \(\chi_{573}(407,\cdot)\) \(\chi_{573}(434,\cdot)\) \(\chi_{573}(503,\cdot)\) \(\chi_{573}(518,\cdot)\) \(\chi_{573}(536,\cdot)\) \(\chi_{573}(542,\cdot)\)

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: \(\Q(\zeta_{19})\)
Fixed field: 38.0.15223168714879313546692095257379448420591016496978036941227483116202289492875331751113046747.1

Values on generators

\((383,19)\) → \((-1,e\left(\frac{11}{19}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 573 }(536, a) \) \(-1\)\(1\)\(e\left(\frac{37}{38}\right)\)\(e\left(\frac{18}{19}\right)\)\(e\left(\frac{17}{38}\right)\)\(1\)\(e\left(\frac{35}{38}\right)\)\(e\left(\frac{8}{19}\right)\)\(e\left(\frac{27}{38}\right)\)\(e\left(\frac{16}{19}\right)\)\(e\left(\frac{37}{38}\right)\)\(e\left(\frac{17}{19}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 573 }(536,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 573 }(536,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 573 }(536,·),\chi_{ 573 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 573 }(536,·)) \;\) at \(\; a,b = \) e.g. 1,2