Properties

Label 9450.3047
Modulus $9450$
Conductor $4725$
Order $180$
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(180))
 
M = H._module
 
chi = DirichletCharacter(H, M([110,153,60]))
 
pari: [g,chi] = znchar(Mod(3047,9450))
 

Basic properties

Modulus: \(9450\)
Conductor: \(4725\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(180\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4725}(3047,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9450.hi

\(\chi_{9450}(23,\cdot)\) \(\chi_{9450}(137,\cdot)\) \(\chi_{9450}(263,\cdot)\) \(\chi_{9450}(527,\cdot)\) \(\chi_{9450}(653,\cdot)\) \(\chi_{9450}(767,\cdot)\) \(\chi_{9450}(1283,\cdot)\) \(\chi_{9450}(1397,\cdot)\) \(\chi_{9450}(1523,\cdot)\) \(\chi_{9450}(1787,\cdot)\) \(\chi_{9450}(1913,\cdot)\) \(\chi_{9450}(2027,\cdot)\) \(\chi_{9450}(2153,\cdot)\) \(\chi_{9450}(2417,\cdot)\) \(\chi_{9450}(2783,\cdot)\) \(\chi_{9450}(3047,\cdot)\) \(\chi_{9450}(3173,\cdot)\) \(\chi_{9450}(3287,\cdot)\) \(\chi_{9450}(3413,\cdot)\) \(\chi_{9450}(3677,\cdot)\) \(\chi_{9450}(3803,\cdot)\) \(\chi_{9450}(3917,\cdot)\) \(\chi_{9450}(4433,\cdot)\) \(\chi_{9450}(4547,\cdot)\) \(\chi_{9450}(4673,\cdot)\) \(\chi_{9450}(4937,\cdot)\) \(\chi_{9450}(5063,\cdot)\) \(\chi_{9450}(5177,\cdot)\) \(\chi_{9450}(5303,\cdot)\) \(\chi_{9450}(5567,\cdot)\) ...

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: $\Q(\zeta_{180})$
Fixed field: Number field defined by a degree 180 polynomial (not computed)

Values on generators

\((9101,6427,6751)\) → \((e\left(\frac{11}{18}\right),e\left(\frac{17}{20}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9450 }(3047, a) \) \(1\)\(1\)\(e\left(\frac{79}{90}\right)\)\(e\left(\frac{7}{180}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{133}{180}\right)\)\(e\left(\frac{14}{45}\right)\)\(e\left(\frac{16}{45}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{71}{90}\right)\)\(e\left(\frac{7}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9450 }(3047,a) \;\) at \(\;a = \) e.g. 2