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Properties

Label 539.141
Modulus $539$
Conductor $539$
Order $35$
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(539, base_ring=CyclotomicField(70))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,42]))
 
pari: [g,chi] = znchar(Mod(141,539))
 

Basic properties

Modulus: \(539\)
Conductor: \(539\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(35\)
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 539.v

\(\chi_{539}(15,\cdot)\) \(\chi_{539}(36,\cdot)\) \(\chi_{539}(64,\cdot)\) \(\chi_{539}(71,\cdot)\) \(\chi_{539}(92,\cdot)\) \(\chi_{539}(113,\cdot)\) \(\chi_{539}(141,\cdot)\) \(\chi_{539}(169,\cdot)\) \(\chi_{539}(190,\cdot)\) \(\chi_{539}(218,\cdot)\) \(\chi_{539}(225,\cdot)\) \(\chi_{539}(267,\cdot)\) \(\chi_{539}(302,\cdot)\) \(\chi_{539}(323,\cdot)\) \(\chi_{539}(372,\cdot)\) \(\chi_{539}(379,\cdot)\) \(\chi_{539}(400,\cdot)\) \(\chi_{539}(421,\cdot)\) \(\chi_{539}(449,\cdot)\) \(\chi_{539}(456,\cdot)\) \(\chi_{539}(477,\cdot)\) \(\chi_{539}(498,\cdot)\) \(\chi_{539}(526,\cdot)\) \(\chi_{539}(533,\cdot)\)

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

Related number fields

Field of values: $\Q(\zeta_{35})$
Fixed field: 35.35.73261800077965937220382205398471606200231960977600836588587975360331959691780081.1

Values on generators

\((199,442)\) → \((e\left(\frac{1}{7}\right),e\left(\frac{3}{5}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\(1\)\(1\)\(e\left(\frac{11}{35}\right)\)\(e\left(\frac{33}{35}\right)\)\(e\left(\frac{22}{35}\right)\)\(e\left(\frac{19}{35}\right)\)\(e\left(\frac{9}{35}\right)\)\(e\left(\frac{33}{35}\right)\)\(e\left(\frac{31}{35}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{11}{35}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 539 }(141,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{539}(141,\cdot)) = \sum_{r\in \Z/539\Z} \chi_{539}(141,r) e\left(\frac{2r}{539}\right) = -15.1470710516+-17.5944945525i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 539 }(141,·),\chi_{ 539 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{539}(141,\cdot),\chi_{539}(1,\cdot)) = \sum_{r\in \Z/539\Z} \chi_{539}(141,r) \chi_{539}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 539 }(141,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{539}(141,·)) = \sum_{r \in \Z/539\Z} \chi_{539}(141,r) e\left(\frac{1 r + 2 r^{-1}}{539}\right) = 0.0 \)