Properties

Label 207.142
Modulus $207$
Conductor $207$
Order $33$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([44,12]))
 
pari: [g,chi] = znchar(Mod(142,207))
 

Basic properties

Modulus: \(207\)
Conductor: \(207\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
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 207.m

\(\chi_{207}(4,\cdot)\) \(\chi_{207}(13,\cdot)\) \(\chi_{207}(16,\cdot)\) \(\chi_{207}(25,\cdot)\) \(\chi_{207}(31,\cdot)\) \(\chi_{207}(49,\cdot)\) \(\chi_{207}(52,\cdot)\) \(\chi_{207}(58,\cdot)\) \(\chi_{207}(85,\cdot)\) \(\chi_{207}(94,\cdot)\) \(\chi_{207}(121,\cdot)\) \(\chi_{207}(124,\cdot)\) \(\chi_{207}(133,\cdot)\) \(\chi_{207}(142,\cdot)\) \(\chi_{207}(151,\cdot)\) \(\chi_{207}(169,\cdot)\) \(\chi_{207}(187,\cdot)\) \(\chi_{207}(193,\cdot)\) \(\chi_{207}(196,\cdot)\) \(\chi_{207}(202,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: 33.33.70011645999218458416472683122408534303895571350166174758601569.1

Values on generators

\((47,28)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{2}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 207 }(142, a) \) \(1\)\(1\)\(e\left(\frac{1}{33}\right)\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{5}{33}\right)\)\(e\left(\frac{4}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 207 }(142,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 207 }(142,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 207 }(142,·),\chi_{ 207 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 207 }(142,·)) \;\) at \(\; a,b = \) e.g. 1,2