Properties

Label 1800.1117
Modulus $1800$
Conductor $200$
Order $20$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,10,0,13]))
 
pari: [g,chi] = znchar(Mod(1117,1800))
 

Basic properties

Modulus: \(1800\)
Conductor: \(200\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{200}(117,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1800.cr

\(\chi_{1800}(37,\cdot)\) \(\chi_{1800}(253,\cdot)\) \(\chi_{1800}(397,\cdot)\) \(\chi_{1800}(613,\cdot)\) \(\chi_{1800}(973,\cdot)\) \(\chi_{1800}(1117,\cdot)\) \(\chi_{1800}(1333,\cdot)\) \(\chi_{1800}(1477,\cdot)\)

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: 20.0.3125000000000000000000000000000000.1

Values on generators

\((1351,901,1001,577)\) → \((1,-1,1,e\left(\frac{13}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1800 }(1117, a) \) \(-1\)\(1\)\(i\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{3}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1800 }(1117,a) \;\) at \(\;a = \) e.g. 2