Properties

Label 667.447
Modulus $667$
Conductor $667$
Order $44$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(667, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,11]))
 
pari: [g,chi] = znchar(Mod(447,667))
 

Basic properties

Modulus: \(667\)
Conductor: \(667\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 667.q

\(\chi_{667}(17,\cdot)\) \(\chi_{667}(99,\cdot)\) \(\chi_{667}(157,\cdot)\) \(\chi_{667}(191,\cdot)\) \(\chi_{667}(244,\cdot)\) \(\chi_{667}(249,\cdot)\) \(\chi_{667}(273,\cdot)\) \(\chi_{667}(336,\cdot)\) \(\chi_{667}(360,\cdot)\) \(\chi_{667}(365,\cdot)\) \(\chi_{667}(389,\cdot)\) \(\chi_{667}(447,\cdot)\) \(\chi_{667}(452,\cdot)\) \(\chi_{667}(481,\cdot)\) \(\chi_{667}(534,\cdot)\) \(\chi_{667}(539,\cdot)\) \(\chi_{667}(563,\cdot)\) \(\chi_{667}(592,\cdot)\) \(\chi_{667}(626,\cdot)\) \(\chi_{667}(655,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: 44.44.2829456642779506738660199294300931896594438764890376623447387777021003184630861677846958804464951379972781.1

Values on generators

\((465,553)\) → \((e\left(\frac{3}{22}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 667 }(447, a) \) \(1\)\(1\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{19}{44}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{21}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 667 }(447,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 667 }(447,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 667 }(447,·),\chi_{ 667 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 667 }(447,·)) \;\) at \(\; a,b = \) e.g. 1,2