Properties

Label 9450.4489
Modulus $9450$
Conductor $4725$
Order $90$
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(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([80,27,30]))
 
pari: [g,chi] = znchar(Mod(4489,9450))
 

Basic properties

Modulus: \(9450\)
Conductor: \(4725\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4725}(4489,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9450.gn

\(\chi_{9450}(79,\cdot)\) \(\chi_{9450}(319,\cdot)\) \(\chi_{9450}(709,\cdot)\) \(\chi_{9450}(1339,\cdot)\) \(\chi_{9450}(1579,\cdot)\) \(\chi_{9450}(1969,\cdot)\) \(\chi_{9450}(2209,\cdot)\) \(\chi_{9450}(2839,\cdot)\) \(\chi_{9450}(3229,\cdot)\) \(\chi_{9450}(3469,\cdot)\) \(\chi_{9450}(3859,\cdot)\) \(\chi_{9450}(4489,\cdot)\) \(\chi_{9450}(4729,\cdot)\) \(\chi_{9450}(5119,\cdot)\) \(\chi_{9450}(5359,\cdot)\) \(\chi_{9450}(5989,\cdot)\) \(\chi_{9450}(6379,\cdot)\) \(\chi_{9450}(6619,\cdot)\) \(\chi_{9450}(7009,\cdot)\) \(\chi_{9450}(7639,\cdot)\) \(\chi_{9450}(7879,\cdot)\) \(\chi_{9450}(8269,\cdot)\) \(\chi_{9450}(8509,\cdot)\) \(\chi_{9450}(9139,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((9101,6427,6751)\) → \((e\left(\frac{8}{9}\right),e\left(\frac{3}{10}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9450 }(4489, a) \) \(1\)\(1\)\(e\left(\frac{31}{45}\right)\)\(e\left(\frac{73}{90}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{67}{90}\right)\)\(e\left(\frac{22}{45}\right)\)\(e\left(\frac{23}{45}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{14}{45}\right)\)\(e\left(\frac{1}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9450 }(4489,a) \;\) at \(\;a = \) e.g. 2