Properties

Label 547.24
Modulus $547$
Conductor $547$
Order $91$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(547, base_ring=CyclotomicField(182))
 
M = H._module
 
chi = DirichletCharacter(H, M([144]))
 
pari: [g,chi] = znchar(Mod(24,547))
 

Basic properties

Modulus: \(547\)
Conductor: \(547\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(91\)
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 547.m

\(\chi_{547}(10,\cdot)\) \(\chi_{547}(24,\cdot)\) \(\chi_{547}(29,\cdot)\) \(\chi_{547}(35,\cdot)\) \(\chi_{547}(44,\cdot)\) \(\chi_{547}(52,\cdot)\) \(\chi_{547}(64,\cdot)\) \(\chi_{547}(84,\cdot)\) \(\chi_{547}(85,\cdot)\) \(\chi_{547}(90,\cdot)\) \(\chi_{547}(93,\cdot)\) \(\chi_{547}(100,\cdot)\) \(\chi_{547}(114,\cdot)\) \(\chi_{547}(131,\cdot)\) \(\chi_{547}(149,\cdot)\) \(\chi_{547}(154,\cdot)\) \(\chi_{547}(161,\cdot)\) \(\chi_{547}(165,\cdot)\) \(\chi_{547}(167,\cdot)\) \(\chi_{547}(179,\cdot)\) \(\chi_{547}(185,\cdot)\) \(\chi_{547}(195,\cdot)\) \(\chi_{547}(204,\cdot)\) \(\chi_{547}(205,\cdot)\) \(\chi_{547}(209,\cdot)\) \(\chi_{547}(212,\cdot)\) \(\chi_{547}(215,\cdot)\) \(\chi_{547}(216,\cdot)\) \(\chi_{547}(218,\cdot)\) \(\chi_{547}(224,\cdot)\) ...

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: $\Q(\zeta_{91})$
Fixed field: Number field defined by a degree 91 polynomial

Values on generators

\(2\) → \(e\left(\frac{72}{91}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 547 }(24, a) \) \(1\)\(1\)\(e\left(\frac{72}{91}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{53}{91}\right)\)\(e\left(\frac{57}{91}\right)\)\(e\left(\frac{20}{91}\right)\)\(e\left(\frac{45}{91}\right)\)\(e\left(\frac{34}{91}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{38}{91}\right)\)\(e\left(\frac{5}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 547 }(24,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 547 }(24,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 547 }(24,·),\chi_{ 547 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 547 }(24,·)) \;\) at \(\; a,b = \) e.g. 1,2