Properties

Label 547.26
Modulus $547$
Conductor $547$
Order $78$
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([41]))
 
pari: [g,chi] = znchar(Mod(26,547))
 

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

Values on generators

\(2\) → \(e\left(\frac{41}{78}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{41}{78}\right)\)\(-1\)\(e\left(\frac{2}{39}\right)\)\(e\left(\frac{7}{78}\right)\)\(e\left(\frac{1}{39}\right)\)\(e\left(\frac{11}{78}\right)\)\(e\left(\frac{15}{26}\right)\)\(1\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{31}{39}\right)\)
value at e.g. 2

Related number fields

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 547 }(26,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{547}(26,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(26,r) e\left(\frac{2r}{547}\right) = 22.3929038281+-6.7496561501i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 547 }(26,·),\chi_{ 547 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{547}(26,\cdot),\chi_{547}(1,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(26,r) \chi_{547}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 547 }(26,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{547}(26,·)) = \sum_{r \in \Z/547\Z} \chi_{547}(26,r) e\left(\frac{1 r + 2 r^{-1}}{547}\right) = -29.6337341229+-2.3922823376i \)