Properties

Label 547.28
Modulus $547$
Conductor $547$
Order $26$
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([5]))
 
pari: [g,chi] = znchar(Mod(28,547))
 

Basic properties

Modulus: \(547\)
Conductor: \(547\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(26\)
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.i

\(\chi_{547}(28,\cdot)\) \(\chi_{547}(30,\cdot)\) \(\chi_{547}(38,\cdot)\) \(\chi_{547}(72,\cdot)\) \(\chi_{547}(107,\cdot)\) \(\chi_{547}(172,\cdot)\) \(\chi_{547}(194,\cdot)\) \(\chi_{547}(197,\cdot)\) \(\chi_{547}(254,\cdot)\) \(\chi_{547}(286,\cdot)\) \(\chi_{547}(310,\cdot)\) \(\chi_{547}(501,\cdot)\)

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

\(2\) → \(e\left(\frac{5}{26}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{5}{26}\right)\)\(-1\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{11}{26}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{21}{26}\right)\)\(e\left(\frac{15}{26}\right)\)\(1\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{6}{13}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{13})\)
Fixed field: 26.0.281632751606716598824960266218413010492516000978852845232181437754307.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 547 }(28,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{547}(28,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(28,r) e\left(\frac{2r}{547}\right) = -5.8651150263+-22.64068077i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 547 }(28,·),\chi_{ 547 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{547}(28,\cdot),\chi_{547}(1,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(28,r) \chi_{547}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 547 }(28,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{547}(28,·)) = \sum_{r \in \Z/547\Z} \chi_{547}(28,r) e\left(\frac{1 r + 2 r^{-1}}{547}\right) = 17.0755499375+-24.7382048977i \)