Properties

Label 1440.19
Modulus $1440$
Conductor $160$
Order $8$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1440, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([4,7,0,4]))
 
pari: [g,chi] = znchar(Mod(19,1440))
 

Basic properties

Modulus: \(1440\)
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 \(\chi_{160}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1440.cm

\(\chi_{1440}(19,\cdot)\) \(\chi_{1440}(379,\cdot)\) \(\chi_{1440}(739,\cdot)\) \(\chi_{1440}(1099,\cdot)\)

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: 8.0.1342177280000.1

Values on generators

\((991,901,641,577)\) → \((-1,e\left(\frac{7}{8}\right),1,-1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1440 }(19, a) \) \(-1\)\(1\)\(-i\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(1\)\(e\left(\frac{5}{8}\right)\)\(i\)\(e\left(\frac{5}{8}\right)\)\(-1\)\(e\left(\frac{3}{8}\right)\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1440 }(19,a) \;\) at \(\;a = \) e.g. 2