Properties

Label 572.525
Modulus $572$
Conductor $143$
Order $20$
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(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,6,15]))
 
pari: [g,chi] = znchar(Mod(525,572))
 

Basic properties

Modulus: \(572\)
Conductor: \(143\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{143}(96,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 572.bh

\(\chi_{572}(57,\cdot)\) \(\chi_{572}(73,\cdot)\) \(\chi_{572}(161,\cdot)\) \(\chi_{572}(281,\cdot)\) \(\chi_{572}(369,\cdot)\) \(\chi_{572}(437,\cdot)\) \(\chi_{572}(525,\cdot)\) \(\chi_{572}(541,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: 20.20.284589332775604260722209388186521117.1

Values on generators

\((287,365,353)\) → \((1,e\left(\frac{3}{10}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(15\)\(17\)\(19\)\(21\)\(23\)\(25\)
\( \chi_{ 572 }(525, a) \) \(1\)\(1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{13}{20}\right)\)\(-i\)\(-1\)\(e\left(\frac{9}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 572 }(525,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 572 }(525,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 572 }(525,·),\chi_{ 572 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 572 }(525,·)) \;\) at \(\; a,b = \) e.g. 1,2