Properties

Label 547.21
Modulus $547$
Conductor $547$
Order $39$
Real no
Primitive yes
Minimal yes
Parity even

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([38]))
 
pari: [g,chi] = znchar(Mod(21,547))
 

Basic properties

Modulus: \(547\)
Conductor: \(547\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(39\)
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.j

\(\chi_{547}(11,\cdot)\) \(\chi_{547}(21,\cdot)\) \(\chi_{547}(47,\cdot)\) \(\chi_{547}(54,\cdot)\) \(\chi_{547}(96,\cdot)\) \(\chi_{547}(121,\cdot)\) \(\chi_{547}(129,\cdot)\) \(\chi_{547}(136,\cdot)\) \(\chi_{547}(181,\cdot)\) \(\chi_{547}(199,\cdot)\) \(\chi_{547}(217,\cdot)\) \(\chi_{547}(231,\cdot)\) \(\chi_{547}(233,\cdot)\) \(\chi_{547}(239,\cdot)\) \(\chi_{547}(296,\cdot)\) \(\chi_{547}(302,\cdot)\) \(\chi_{547}(325,\cdot)\) \(\chi_{547}(402,\cdot)\) \(\chi_{547}(419,\cdot)\) \(\chi_{547}(441,\cdot)\) \(\chi_{547}(445,\cdot)\) \(\chi_{547}(464,\cdot)\) \(\chi_{547}(488,\cdot)\) \(\chi_{547}(521,\cdot)\)

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

\(2\) → \(e\left(\frac{38}{39}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{38}{39}\right)\)\(1\)\(e\left(\frac{37}{39}\right)\)\(e\left(\frac{16}{39}\right)\)\(e\left(\frac{38}{39}\right)\)\(e\left(\frac{14}{39}\right)\)\(e\left(\frac{12}{13}\right)\)\(1\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{8}{39}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{39})$
Fixed field: 39.39.110539730308983252644692019573737637228828219789892386533857010792412648335681998371651796263456819412889.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 547 }(21,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{547}(21,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(21,r) e\left(\frac{2r}{547}\right) = 18.5670653063+14.2219578789i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 547 }(21,·),\chi_{ 547 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{547}(21,\cdot),\chi_{547}(1,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(21,r) \chi_{547}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 547 }(21,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{547}(21,·)) = \sum_{r \in \Z/547\Z} \chi_{547}(21,r) e\left(\frac{1 r + 2 r^{-1}}{547}\right) = -38.9879150672+3.1474298922i \)