Properties

Label 1078.17
Modulus $1078$
Conductor $539$
Order $210$
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([125,189]))
 
pari: [g,chi] = znchar(Mod(17,1078))
 

Basic properties

Modulus: \(1078\)
Conductor: \(539\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(210\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{539}(17,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1078.bf

\(\chi_{1078}(17,\cdot)\) \(\chi_{1078}(61,\cdot)\) \(\chi_{1078}(73,\cdot)\) \(\chi_{1078}(101,\cdot)\) \(\chi_{1078}(145,\cdot)\) \(\chi_{1078}(171,\cdot)\) \(\chi_{1078}(173,\cdot)\) \(\chi_{1078}(255,\cdot)\) \(\chi_{1078}(271,\cdot)\) \(\chi_{1078}(283,\cdot)\) \(\chi_{1078}(299,\cdot)\) \(\chi_{1078}(327,\cdot)\) \(\chi_{1078}(369,\cdot)\) \(\chi_{1078}(381,\cdot)\) \(\chi_{1078}(409,\cdot)\) \(\chi_{1078}(425,\cdot)\) \(\chi_{1078}(437,\cdot)\) \(\chi_{1078}(453,\cdot)\) \(\chi_{1078}(479,\cdot)\) \(\chi_{1078}(481,\cdot)\) \(\chi_{1078}(523,\cdot)\) \(\chi_{1078}(535,\cdot)\) \(\chi_{1078}(563,\cdot)\) \(\chi_{1078}(579,\cdot)\) \(\chi_{1078}(591,\cdot)\) \(\chi_{1078}(633,\cdot)\) \(\chi_{1078}(635,\cdot)\) \(\chi_{1078}(677,\cdot)\) \(\chi_{1078}(689,\cdot)\) \(\chi_{1078}(733,\cdot)\) ...

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 210 polynomial (not computed)

Values on generators

\((199,981)\) → \((e\left(\frac{25}{42}\right),e\left(\frac{9}{10}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\(1\)\(1\)\(e\left(\frac{167}{210}\right)\)\(e\left(\frac{181}{210}\right)\)\(e\left(\frac{62}{105}\right)\)\(e\left(\frac{19}{35}\right)\)\(e\left(\frac{23}{35}\right)\)\(e\left(\frac{103}{105}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{76}{105}\right)\)\(e\left(\frac{27}{70}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1078 }(17,a) \;\) at \(\;a = \) e.g. 2