Properties

Label 9450.3251
Modulus $9450$
Conductor $189$
Order $18$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9450, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([13,0,3]))
 
pari: [g,chi] = znchar(Mod(3251,9450))
 

Basic properties

Modulus: \(9450\)
Conductor: \(189\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{189}(38,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9450.dp

\(\chi_{9450}(101,\cdot)\) \(\chi_{9450}(2651,\cdot)\) \(\chi_{9450}(3251,\cdot)\) \(\chi_{9450}(5801,\cdot)\) \(\chi_{9450}(6401,\cdot)\) \(\chi_{9450}(8951,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: Number field defined by a degree 18 polynomial

Values on generators

\((9101,6427,6751)\) → \((e\left(\frac{13}{18}\right),1,e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9450 }(3251, a) \) \(1\)\(1\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{5}{18}\right)\)\(1\)\(-1\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{8}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9450 }(3251,a) \;\) at \(\;a = \) e.g. 2