Properties

Label 147.44
Modulus $147$
Conductor $147$
Order $42$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,8]))
 
pari: [g,chi] = znchar(Mod(44,147))
 

Basic properties

Modulus: \(147\)
Conductor: \(147\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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 147.n

\(\chi_{147}(2,\cdot)\) \(\chi_{147}(11,\cdot)\) \(\chi_{147}(23,\cdot)\) \(\chi_{147}(32,\cdot)\) \(\chi_{147}(44,\cdot)\) \(\chi_{147}(53,\cdot)\) \(\chi_{147}(65,\cdot)\) \(\chi_{147}(74,\cdot)\) \(\chi_{147}(86,\cdot)\) \(\chi_{147}(95,\cdot)\) \(\chi_{147}(107,\cdot)\) \(\chi_{147}(137,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: 42.0.176602720807616761537805583365440112858555316650025456145851095351761290003.1

Values on generators

\((50,52)\) → \((-1,e\left(\frac{4}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 147 }(44, a) \) \(-1\)\(1\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 147 }(44,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 147 }(44,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 147 }(44,·),\chi_{ 147 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 147 }(44,·)) \;\) at \(\; a,b = \) e.g. 1,2