Properties

Label 9450.5171
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([85,54,75]))
 
pari: [g,chi] = znchar(Mod(5171,9450))
 

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 \(\chi_{4725}(446,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9450.gf

\(\chi_{9450}(131,\cdot)\) \(\chi_{9450}(731,\cdot)\) \(\chi_{9450}(761,\cdot)\) \(\chi_{9450}(1361,\cdot)\) \(\chi_{9450}(1391,\cdot)\) \(\chi_{9450}(1991,\cdot)\) \(\chi_{9450}(2021,\cdot)\) \(\chi_{9450}(2621,\cdot)\) \(\chi_{9450}(3281,\cdot)\) \(\chi_{9450}(3881,\cdot)\) \(\chi_{9450}(3911,\cdot)\) \(\chi_{9450}(4511,\cdot)\) \(\chi_{9450}(4541,\cdot)\) \(\chi_{9450}(5141,\cdot)\) \(\chi_{9450}(5171,\cdot)\) \(\chi_{9450}(5771,\cdot)\) \(\chi_{9450}(6431,\cdot)\) \(\chi_{9450}(7031,\cdot)\) \(\chi_{9450}(7061,\cdot)\) \(\chi_{9450}(7661,\cdot)\) \(\chi_{9450}(7691,\cdot)\) \(\chi_{9450}(8291,\cdot)\) \(\chi_{9450}(8321,\cdot)\) \(\chi_{9450}(8921,\cdot)\)

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

\((9101,6427,6751)\) → \((e\left(\frac{17}{18}\right),e\left(\frac{3}{5}\right),e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9450 }(5171, a) \) \(1\)\(1\)\(e\left(\frac{19}{90}\right)\)\(e\left(\frac{41}{90}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{59}{90}\right)\)\(e\left(\frac{13}{90}\right)\)\(e\left(\frac{47}{90}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{43}{45}\right)\)\(e\left(\frac{7}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9450 }(5171,a) \;\) at \(\;a = \) e.g. 2