Properties

Label 9450.2887
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([100,81,30]))
 
pari: [g,chi] = znchar(Mod(2887,9450))
 

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 \(\chi_{4725}(2887,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9450.gz

\(\chi_{9450}(103,\cdot)\) \(\chi_{9450}(367,\cdot)\) \(\chi_{9450}(733,\cdot)\) \(\chi_{9450}(997,\cdot)\) \(\chi_{9450}(1123,\cdot)\) \(\chi_{9450}(1237,\cdot)\) \(\chi_{9450}(1363,\cdot)\) \(\chi_{9450}(1627,\cdot)\) \(\chi_{9450}(1753,\cdot)\) \(\chi_{9450}(1867,\cdot)\) \(\chi_{9450}(2383,\cdot)\) \(\chi_{9450}(2497,\cdot)\) \(\chi_{9450}(2623,\cdot)\) \(\chi_{9450}(2887,\cdot)\) \(\chi_{9450}(3013,\cdot)\) \(\chi_{9450}(3127,\cdot)\) \(\chi_{9450}(3253,\cdot)\) \(\chi_{9450}(3517,\cdot)\) \(\chi_{9450}(3883,\cdot)\) \(\chi_{9450}(4147,\cdot)\) \(\chi_{9450}(4273,\cdot)\) \(\chi_{9450}(4387,\cdot)\) \(\chi_{9450}(4513,\cdot)\) \(\chi_{9450}(4777,\cdot)\) \(\chi_{9450}(4903,\cdot)\) \(\chi_{9450}(5017,\cdot)\) \(\chi_{9450}(5533,\cdot)\) \(\chi_{9450}(5647,\cdot)\) \(\chi_{9450}(5773,\cdot)\) \(\chi_{9450}(6037,\cdot)\) ...

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

\((9101,6427,6751)\) → \((e\left(\frac{5}{9}\right),e\left(\frac{9}{20}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9450 }(2887, a) \) \(1\)\(1\)\(e\left(\frac{4}{45}\right)\)\(e\left(\frac{89}{180}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{71}{180}\right)\)\(e\left(\frac{41}{90}\right)\)\(e\left(\frac{79}{90}\right)\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{67}{90}\right)\)\(e\left(\frac{35}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9450 }(2887,a) \;\) at \(\;a = \) e.g. 2