Properties

Label 609.326
Modulus $609$
Conductor $609$
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(609, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,28,18]))
 
pari: [g,chi] = znchar(Mod(326,609))
 

Basic properties

Modulus: \(609\)
Conductor: \(609\)
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 609.bl

\(\chi_{609}(23,\cdot)\) \(\chi_{609}(53,\cdot)\) \(\chi_{609}(65,\cdot)\) \(\chi_{609}(74,\cdot)\) \(\chi_{609}(107,\cdot)\) \(\chi_{609}(170,\cdot)\) \(\chi_{609}(284,\cdot)\) \(\chi_{609}(326,\cdot)\) \(\chi_{609}(368,\cdot)\) \(\chi_{609}(401,\cdot)\) \(\chi_{609}(422,\cdot)\) \(\chi_{609}(431,\cdot)\)

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: Number field defined by a degree 42 polynomial

Values on generators

\((407,262,379)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{3}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 609 }(326, a) \) \(-1\)\(1\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{4}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 609 }(326,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 609 }(326,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 609 }(326,·),\chi_{ 609 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 609 }(326,·)) \;\) at \(\; a,b = \) e.g. 1,2