Properties

Label 2240.169
Modulus $2240$
Conductor $160$
Order $8$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,7,4,0]))
 
pari: [g,chi] = znchar(Mod(169,2240))
 

Basic properties

Modulus: \(2240\)
Conductor: \(160\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{160}(109,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2240.ct

\(\chi_{2240}(169,\cdot)\) \(\chi_{2240}(729,\cdot)\) \(\chi_{2240}(1289,\cdot)\) \(\chi_{2240}(1849,\cdot)\)

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: 8.8.1342177280000.1

Values on generators

\((1471,1541,897,1921)\) → \((1,e\left(\frac{7}{8}\right),-1,1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 2240 }(169, a) \) \(1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(i\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(1\)\(e\left(\frac{1}{8}\right)\)\(-i\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2240 }(169,a) \;\) at \(\;a = \) e.g. 2