Properties

Label 1078.9
Modulus $1078$
Conductor $539$
Order $105$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1078, base_ring=CyclotomicField(210))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,126]))
 
pari: [g,chi] = znchar(Mod(9,1078))
 

Basic properties

Modulus: \(1078\)
Conductor: \(539\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(105\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{539}(9,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1078.bc

\(\chi_{1078}(9,\cdot)\) \(\chi_{1078}(25,\cdot)\) \(\chi_{1078}(37,\cdot)\) \(\chi_{1078}(53,\cdot)\) \(\chi_{1078}(81,\cdot)\) \(\chi_{1078}(93,\cdot)\) \(\chi_{1078}(135,\cdot)\) \(\chi_{1078}(137,\cdot)\) \(\chi_{1078}(163,\cdot)\) \(\chi_{1078}(179,\cdot)\) \(\chi_{1078}(191,\cdot)\) \(\chi_{1078}(207,\cdot)\) \(\chi_{1078}(235,\cdot)\) \(\chi_{1078}(247,\cdot)\) \(\chi_{1078}(289,\cdot)\) \(\chi_{1078}(291,\cdot)\) \(\chi_{1078}(317,\cdot)\) \(\chi_{1078}(333,\cdot)\) \(\chi_{1078}(345,\cdot)\) \(\chi_{1078}(389,\cdot)\) \(\chi_{1078}(401,\cdot)\) \(\chi_{1078}(443,\cdot)\) \(\chi_{1078}(445,\cdot)\) \(\chi_{1078}(487,\cdot)\) \(\chi_{1078}(499,\cdot)\) \(\chi_{1078}(515,\cdot)\) \(\chi_{1078}(543,\cdot)\) \(\chi_{1078}(555,\cdot)\) \(\chi_{1078}(597,\cdot)\) \(\chi_{1078}(599,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 105 polynomial (not computed)

Values on generators

\((199,981)\) → \((e\left(\frac{1}{21}\right),e\left(\frac{3}{5}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\(1\)\(1\)\(e\left(\frac{89}{105}\right)\)\(e\left(\frac{82}{105}\right)\)\(e\left(\frac{73}{105}\right)\)\(e\left(\frac{6}{35}\right)\)\(e\left(\frac{22}{35}\right)\)\(e\left(\frac{62}{105}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{59}{105}\right)\)\(e\left(\frac{19}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1078 }(9,a) \;\) at \(\;a = \) e.g. 2