Properties

Label 552.163
Modulus $552$
Conductor $184$
Order $22$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(552, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,11,0,2]))
 
pari: [g,chi] = znchar(Mod(163,552))
 

Basic properties

Modulus: \(552\)
Conductor: \(184\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{184}(163,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 552.bd

\(\chi_{552}(163,\cdot)\) \(\chi_{552}(187,\cdot)\) \(\chi_{552}(211,\cdot)\) \(\chi_{552}(259,\cdot)\) \(\chi_{552}(307,\cdot)\) \(\chi_{552}(331,\cdot)\) \(\chi_{552}(403,\cdot)\) \(\chi_{552}(427,\cdot)\) \(\chi_{552}(499,\cdot)\) \(\chi_{552}(547,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: 22.0.14741666340843480753092741810452692992.1

Values on generators

\((415,277,185,97)\) → \((-1,-1,1,e\left(\frac{1}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 552 }(163, a) \) \(-1\)\(1\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{9}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 552 }(163,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 552 }(163,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 552 }(163,·),\chi_{ 552 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 552 }(163,·)) \;\) at \(\; a,b = \) e.g. 1,2