Properties

Label 572.391
Modulus $572$
Conductor $44$
Order $10$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(572, base_ring=CyclotomicField(10))
 
M = H._module
 
chi = DirichletCharacter(H, M([5,9,0]))
 
pari: [g,chi] = znchar(Mod(391,572))
 

Basic properties

Modulus: \(572\)
Conductor: \(44\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(10\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{44}(39,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 572.y

\(\chi_{572}(79,\cdot)\) \(\chi_{572}(183,\cdot)\) \(\chi_{572}(391,\cdot)\) \(\chi_{572}(547,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: \(\Q(\zeta_{44})^+\)

Values on generators

\((287,365,353)\) → \((-1,e\left(\frac{9}{10}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(15\)\(17\)\(19\)\(21\)\(23\)\(25\)
\( \chi_{ 572 }(391, a) \) \(1\)\(1\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(-1\)\(-1\)\(e\left(\frac{1}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 572 }(391,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 572 }(391,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 572 }(391,·),\chi_{ 572 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 572 }(391,·)) \;\) at \(\; a,b = \) e.g. 1,2