Properties

Label 1800.1367
Modulus $1800$
Conductor $300$
Order $20$
Real no
Primitive no
Minimal no
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([10,0,10,13]))
 
pari: [g,chi] = znchar(Mod(1367,1800))
 

Basic properties

Modulus: \(1800\)
Conductor: \(300\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{300}(167,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1800.cn

\(\chi_{1800}(287,\cdot)\) \(\chi_{1800}(503,\cdot)\) \(\chi_{1800}(647,\cdot)\) \(\chi_{1800}(863,\cdot)\) \(\chi_{1800}(1223,\cdot)\) \(\chi_{1800}(1367,\cdot)\) \(\chi_{1800}(1583,\cdot)\) \(\chi_{1800}(1727,\cdot)\)

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: 20.0.180203247070312500000000000000000000.1

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1800 }(1367, a) \) \(-1\)\(1\)\(-i\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{1}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1800 }(1367,a) \;\) at \(\;a = \) e.g. 2