Properties

Label 3960.163
Modulus $3960$
Conductor $440$
Order $20$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3960, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,10,0,15,12]))
 
pari: [g,chi] = znchar(Mod(163,3960))
 

Basic properties

Modulus: \(3960\)
Conductor: \(440\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{440}(163,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3960.fj

\(\chi_{3960}(163,\cdot)\) \(\chi_{3960}(883,\cdot)\) \(\chi_{3960}(1027,\cdot)\) \(\chi_{3960}(1747,\cdot)\) \(\chi_{3960}(1963,\cdot)\) \(\chi_{3960}(2467,\cdot)\) \(\chi_{3960}(3403,\cdot)\) \(\chi_{3960}(3547,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: Number field defined by a degree 20 polynomial

Values on generators

\((991,1981,3521,2377,2521)\) → \((-1,-1,1,-i,e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 3960 }(163, a) \) \(1\)\(1\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(-i\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3960 }(163,a) \;\) at \(\;a = \) e.g. 2