Properties

Label 209.8
Modulus $209$
Conductor $209$
Order $30$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(209, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([9,5]))
 
pari: [g,chi] = znchar(Mod(8,209))
 

Basic properties

Modulus: \(209\)
Conductor: \(209\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
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 209.t

\(\chi_{209}(8,\cdot)\) \(\chi_{209}(46,\cdot)\) \(\chi_{209}(50,\cdot)\) \(\chi_{209}(84,\cdot)\) \(\chi_{209}(107,\cdot)\) \(\chi_{209}(145,\cdot)\) \(\chi_{209}(160,\cdot)\) \(\chi_{209}(183,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 30.30.1220232317838205647399552173000517992590495082743137882605129.1

Values on generators

\((134,78)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 209 }(8, a) \) \(1\)\(1\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{1}{3}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 209 }(8,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 209 }(8,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 209 }(8,·),\chi_{ 209 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 209 }(8,·)) \;\) at \(\; a,b = \) e.g. 1,2