Properties

Label 547.5
Modulus $547$
Conductor $547$
Order $546$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(547)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([179]))
 
pari: [g,chi] = znchar(Mod(5,547))
 

Basic properties

Modulus: \(547\)
Conductor: \(547\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(546\)
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.p

\(\chi_{547}(2,\cdot)\) \(\chi_{547}(5,\cdot)\) \(\chi_{547}(7,\cdot)\) \(\chi_{547}(12,\cdot)\) \(\chi_{547}(17,\cdot)\) \(\chi_{547}(18,\cdot)\) \(\chi_{547}(20,\cdot)\) \(\chi_{547}(22,\cdot)\) \(\chi_{547}(23,\cdot)\) \(\chi_{547}(32,\cdot)\) \(\chi_{547}(33,\cdot)\) \(\chi_{547}(37,\cdot)\) \(\chi_{547}(43,\cdot)\) \(\chi_{547}(45,\cdot)\) \(\chi_{547}(48,\cdot)\) \(\chi_{547}(50,\cdot)\) \(\chi_{547}(57,\cdot)\) \(\chi_{547}(58,\cdot)\) \(\chi_{547}(61,\cdot)\) \(\chi_{547}(63,\cdot)\) \(\chi_{547}(70,\cdot)\) \(\chi_{547}(71,\cdot)\) \(\chi_{547}(75,\cdot)\) \(\chi_{547}(77,\cdot)\) \(\chi_{547}(88,\cdot)\) \(\chi_{547}(91,\cdot)\) \(\chi_{547}(92,\cdot)\) \(\chi_{547}(95,\cdot)\) \(\chi_{547}(104,\cdot)\) \(\chi_{547}(108,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\(2\) → \(e\left(\frac{179}{546}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{179}{546}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{179}{273}\right)\)\(e\left(\frac{373}{546}\right)\)\(e\left(\frac{265}{273}\right)\)\(e\left(\frac{419}{546}\right)\)\(e\left(\frac{179}{182}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{1}{91}\right)\)\(e\left(\frac{37}{39}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{273})$
Fixed field: Number field defined by a degree 546 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 547 }(5,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{547}(5,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(5,r) e\left(\frac{2r}{547}\right) = 14.7740651045+18.1308301047i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 547 }(5,·),\chi_{ 547 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{547}(5,\cdot),\chi_{547}(1,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(5,r) \chi_{547}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 547 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{547}(5,·)) = \sum_{r \in \Z/547\Z} \chi_{547}(5,r) e\left(\frac{1 r + 2 r^{-1}}{547}\right) = -15.9132508046+9.5574922346i \)