Properties

Label 5390.139
Modulus $5390$
Conductor $2695$
Order $70$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5390, base_ring=CyclotomicField(70))
 
M = H._module
 
chi = DirichletCharacter(H, M([35,25,49]))
 
pari: [g,chi] = znchar(Mod(139,5390))
 

Basic properties

Modulus: \(5390\)
Conductor: \(2695\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(70\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2695}(139,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5390.cv

\(\chi_{5390}(139,\cdot)\) \(\chi_{5390}(349,\cdot)\) \(\chi_{5390}(629,\cdot)\) \(\chi_{5390}(699,\cdot)\) \(\chi_{5390}(909,\cdot)\) \(\chi_{5390}(1119,\cdot)\) \(\chi_{5390}(1399,\cdot)\) \(\chi_{5390}(1679,\cdot)\) \(\chi_{5390}(1889,\cdot)\) \(\chi_{5390}(2169,\cdot)\) \(\chi_{5390}(2239,\cdot)\) \(\chi_{5390}(2659,\cdot)\) \(\chi_{5390}(3009,\cdot)\) \(\chi_{5390}(3219,\cdot)\) \(\chi_{5390}(3709,\cdot)\) \(\chi_{5390}(3779,\cdot)\) \(\chi_{5390}(3989,\cdot)\) \(\chi_{5390}(4199,\cdot)\) \(\chi_{5390}(4479,\cdot)\) \(\chi_{5390}(4549,\cdot)\) \(\chi_{5390}(4759,\cdot)\) \(\chi_{5390}(4969,\cdot)\) \(\chi_{5390}(5249,\cdot)\) \(\chi_{5390}(5319,\cdot)\)

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

Related number fields

Field of values: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 70 polynomial

Values on generators

\((2157,4511,981)\) → \((-1,e\left(\frac{5}{14}\right),e\left(\frac{7}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\( \chi_{ 5390 }(139, a) \) \(1\)\(1\)\(e\left(\frac{16}{35}\right)\)\(e\left(\frac{32}{35}\right)\)\(e\left(\frac{69}{70}\right)\)\(e\left(\frac{51}{70}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{13}{35}\right)\)\(e\left(\frac{23}{70}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{23}{70}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5390 }(139,a) \;\) at \(\;a = \) e.g. 2