Properties

Label 1200.163
Modulus $1200$
Conductor $400$
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(1200, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,15,0,19]))
 
pari: [g,chi] = znchar(Mod(163,1200))
 

Basic properties

Modulus: \(1200\)
Conductor: \(400\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{400}(163,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1200.ct

\(\chi_{1200}(163,\cdot)\) \(\chi_{1200}(187,\cdot)\) \(\chi_{1200}(403,\cdot)\) \(\chi_{1200}(427,\cdot)\) \(\chi_{1200}(667,\cdot)\) \(\chi_{1200}(883,\cdot)\) \(\chi_{1200}(1123,\cdot)\) \(\chi_{1200}(1147,\cdot)\)

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: 20.20.104857600000000000000000000000000000000000.2

Values on generators

\((751,901,401,577)\) → \((-1,-i,1,e\left(\frac{19}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1200 }(163, a) \) \(1\)\(1\)\(-i\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1200 }(163,a) \;\) at \(\;a = \) e.g. 2