Properties

Label 1920.869
Modulus $1920$
Conductor $1920$
Order $32$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1920, base_ring=CyclotomicField(32))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,16,16]))
 
pari: [g,chi] = znchar(Mod(869,1920))
 

Basic properties

Modulus: \(1920\)
Conductor: \(1920\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
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 1920.db

\(\chi_{1920}(29,\cdot)\) \(\chi_{1920}(149,\cdot)\) \(\chi_{1920}(269,\cdot)\) \(\chi_{1920}(389,\cdot)\) \(\chi_{1920}(509,\cdot)\) \(\chi_{1920}(629,\cdot)\) \(\chi_{1920}(749,\cdot)\) \(\chi_{1920}(869,\cdot)\) \(\chi_{1920}(989,\cdot)\) \(\chi_{1920}(1109,\cdot)\) \(\chi_{1920}(1229,\cdot)\) \(\chi_{1920}(1349,\cdot)\) \(\chi_{1920}(1469,\cdot)\) \(\chi_{1920}(1589,\cdot)\) \(\chi_{1920}(1709,\cdot)\) \(\chi_{1920}(1829,\cdot)\)

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: \(\Q(\zeta_{32})\)
Fixed field: 32.0.20615283744186880032112588280912120153429260426382010514145280000000000000000.1

Values on generators

\((511,901,641,1537)\) → \((1,e\left(\frac{9}{32}\right),-1,-1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1920 }(869, a) \) \(-1\)\(1\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{23}{32}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{3}{32}\right)\)\(i\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{15}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1920 }(869,a) \;\) at \(\;a = \) e.g. 2