Properties

Label 3040.1709
Modulus $3040$
Conductor $3040$
Order $8$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,7,4,4]))
 
pari: [g,chi] = znchar(Mod(1709,3040))
 

Basic properties

Modulus: \(3040\)
Conductor: \(3040\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3040.cn

\(\chi_{3040}(189,\cdot)\) \(\chi_{3040}(949,\cdot)\) \(\chi_{3040}(1709,\cdot)\) \(\chi_{3040}(2469,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: 8.0.174913885306880000.1

Values on generators

\((191,2661,1217,1921)\) → \((1,e\left(\frac{7}{8}\right),-1,-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 3040 }(1709, a) \) \(-1\)\(1\)\(e\left(\frac{5}{8}\right)\)\(i\)\(i\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(1\)\(e\left(\frac{7}{8}\right)\)\(-i\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3040 }(1709,a) \;\) at \(\;a = \) e.g. 2