Properties

Label 1078.1055
Modulus $1078$
Conductor $539$
Order $42$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1078, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([17,21]))
 
pari: [g,chi] = znchar(Mod(1055,1078))
 

Basic properties

Modulus: \(1078\)
Conductor: \(539\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{539}(516,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1078.w

\(\chi_{1078}(87,\cdot)\) \(\chi_{1078}(131,\cdot)\) \(\chi_{1078}(241,\cdot)\) \(\chi_{1078}(285,\cdot)\) \(\chi_{1078}(395,\cdot)\) \(\chi_{1078}(439,\cdot)\) \(\chi_{1078}(549,\cdot)\) \(\chi_{1078}(593,\cdot)\) \(\chi_{1078}(703,\cdot)\) \(\chi_{1078}(747,\cdot)\) \(\chi_{1078}(857,\cdot)\) \(\chi_{1078}(1055,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: 42.42.874571799455406731006354153891279294008499270793470105314249037985839659586368645179677.1

Values on generators

\((199,981)\) → \((e\left(\frac{17}{42}\right),-1)\)

Values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 1078 }(1055, a) \) \(1\)\(1\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{3}{14}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1078 }(1055,a) \;\) at \(\;a = \) e.g. 2