Properties

Label 9450.3683
Modulus $9450$
Conductor $675$
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([130,27,0]))
 
pari: [g,chi] = znchar(Mod(3683,9450))
 

Basic properties

Modulus: \(9450\)
Conductor: \(675\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(180\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{675}(308,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9450.hb

\(\chi_{9450}(113,\cdot)\) \(\chi_{9450}(533,\cdot)\) \(\chi_{9450}(617,\cdot)\) \(\chi_{9450}(1037,\cdot)\) \(\chi_{9450}(1163,\cdot)\) \(\chi_{9450}(1247,\cdot)\) \(\chi_{9450}(1373,\cdot)\) \(\chi_{9450}(1667,\cdot)\) \(\chi_{9450}(1877,\cdot)\) \(\chi_{9450}(2003,\cdot)\) \(\chi_{9450}(2297,\cdot)\) \(\chi_{9450}(2423,\cdot)\) \(\chi_{9450}(2633,\cdot)\) \(\chi_{9450}(2927,\cdot)\) \(\chi_{9450}(3053,\cdot)\) \(\chi_{9450}(3137,\cdot)\) \(\chi_{9450}(3263,\cdot)\) \(\chi_{9450}(3683,\cdot)\) \(\chi_{9450}(3767,\cdot)\) \(\chi_{9450}(4187,\cdot)\) \(\chi_{9450}(4313,\cdot)\) \(\chi_{9450}(4397,\cdot)\) \(\chi_{9450}(4523,\cdot)\) \(\chi_{9450}(4817,\cdot)\) \(\chi_{9450}(5027,\cdot)\) \(\chi_{9450}(5153,\cdot)\) \(\chi_{9450}(5447,\cdot)\) \(\chi_{9450}(5573,\cdot)\) \(\chi_{9450}(5783,\cdot)\) \(\chi_{9450}(6077,\cdot)\) ...

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

\((9101,6427,6751)\) → \((e\left(\frac{13}{18}\right),e\left(\frac{3}{20}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9450 }(3683, a) \) \(1\)\(1\)\(e\left(\frac{71}{90}\right)\)\(e\left(\frac{113}{180}\right)\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{107}{180}\right)\)\(e\left(\frac{1}{45}\right)\)\(e\left(\frac{29}{45}\right)\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{79}{90}\right)\)\(e\left(\frac{5}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9450 }(3683,a) \;\) at \(\;a = \) e.g. 2