Properties

Label 9450.4213
Modulus $9450$
Conductor $175$
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(9450, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,19,10]))
 
pari: [g,chi] = znchar(Mod(4213,9450))
 

Basic properties

Modulus: \(9450\)
Conductor: \(175\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{175}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9450.ds

\(\chi_{9450}(433,\cdot)\) \(\chi_{9450}(1567,\cdot)\) \(\chi_{9450}(2323,\cdot)\) \(\chi_{9450}(4213,\cdot)\) \(\chi_{9450}(5347,\cdot)\) \(\chi_{9450}(6103,\cdot)\) \(\chi_{9450}(7237,\cdot)\) \(\chi_{9450}(9127,\cdot)\)

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: 20.20.822111175511963665485382080078125.1

Values on generators

\((9101,6427,6751)\) → \((1,e\left(\frac{19}{20}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9450 }(4213, a) \) \(1\)\(1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9450 }(4213,a) \;\) at \(\;a = \) e.g. 2