Properties

Label 209.69
Modulus $209$
Conductor $209$
Order $30$
Real no
Primitive yes
Minimal yes
Parity odd

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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([24,25]))
 
pari: [g,chi] = znchar(Mod(69,209))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 209.r

\(\chi_{209}(27,\cdot)\) \(\chi_{209}(31,\cdot)\) \(\chi_{209}(69,\cdot)\) \(\chi_{209}(103,\cdot)\) \(\chi_{209}(126,\cdot)\) \(\chi_{209}(141,\cdot)\) \(\chi_{209}(179,\cdot)\) \(\chi_{209}(202,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 30 polynomial

Values on generators

\((134,78)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 209 }(69, a) \) \(-1\)\(1\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{1}{6}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 209 }(69,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 209 }(69,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 209 }(69,·),\chi_{ 209 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 209 }(69,·)) \;\) at \(\; a,b = \) e.g. 1,2