Properties

Label 547.330
Modulus $547$
Conductor $547$
Order $78$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(547, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([19]))
 
pari: [g,chi] = znchar(Mod(330,547))
 

Basic properties

Modulus: \(547\)
Conductor: \(547\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
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 547.l

\(\chi_{547}(26,\cdot)\) \(\chi_{547}(59,\cdot)\) \(\chi_{547}(83,\cdot)\) \(\chi_{547}(102,\cdot)\) \(\chi_{547}(106,\cdot)\) \(\chi_{547}(128,\cdot)\) \(\chi_{547}(145,\cdot)\) \(\chi_{547}(222,\cdot)\) \(\chi_{547}(245,\cdot)\) \(\chi_{547}(251,\cdot)\) \(\chi_{547}(308,\cdot)\) \(\chi_{547}(314,\cdot)\) \(\chi_{547}(316,\cdot)\) \(\chi_{547}(330,\cdot)\) \(\chi_{547}(348,\cdot)\) \(\chi_{547}(366,\cdot)\) \(\chi_{547}(411,\cdot)\) \(\chi_{547}(418,\cdot)\) \(\chi_{547}(426,\cdot)\) \(\chi_{547}(451,\cdot)\) \(\chi_{547}(493,\cdot)\) \(\chi_{547}(500,\cdot)\) \(\chi_{547}(526,\cdot)\) \(\chi_{547}(536,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\(2\) → \(e\left(\frac{19}{78}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 547 }(330, a) \) \(-1\)\(1\)\(e\left(\frac{19}{78}\right)\)\(-1\)\(e\left(\frac{19}{39}\right)\)\(e\left(\frac{47}{78}\right)\)\(e\left(\frac{29}{39}\right)\)\(e\left(\frac{7}{78}\right)\)\(e\left(\frac{19}{26}\right)\)\(1\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{2}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 547 }(330,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 547 }(330,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 547 }(330,·),\chi_{ 547 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 547 }(330,·)) \;\) at \(\; a,b = \) e.g. 1,2