Properties

Label 1078.195
Modulus $1078$
Conductor $77$
Order $10$
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(10))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([5,3]))
 
pari: [g,chi] = znchar(Mod(195,1078))
 

Basic properties

Modulus: \(1078\)
Conductor: \(77\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(10\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{77}(41,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1078.l

\(\chi_{1078}(195,\cdot)\) \(\chi_{1078}(293,\cdot)\) \(\chi_{1078}(391,\cdot)\) \(\chi_{1078}(783,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: 10.10.39630026842637.1

Values on generators

\((199,981)\) → \((-1,e\left(\frac{3}{10}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\(1\)\(1\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{7}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1078 }(195,a) \;\) at \(\;a = \) e.g. 2