Properties

Label 9450.3223
Modulus $9450$
Conductor $1575$
Order $60$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9450, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([20,33,10]))
 
pari: [g,chi] = znchar(Mod(3223,9450))
 

Basic properties

Modulus: \(9450\)
Conductor: \(1575\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1575}(598,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9450.fo

\(\chi_{9450}(397,\cdot)\) \(\chi_{9450}(577,\cdot)\) \(\chi_{9450}(1153,\cdot)\) \(\chi_{9450}(1333,\cdot)\) \(\chi_{9450}(2287,\cdot)\) \(\chi_{9450}(2467,\cdot)\) \(\chi_{9450}(3223,\cdot)\) \(\chi_{9450}(4177,\cdot)\) \(\chi_{9450}(4933,\cdot)\) \(\chi_{9450}(5113,\cdot)\) \(\chi_{9450}(6067,\cdot)\) \(\chi_{9450}(6247,\cdot)\) \(\chi_{9450}(6823,\cdot)\) \(\chi_{9450}(7003,\cdot)\) \(\chi_{9450}(8137,\cdot)\) \(\chi_{9450}(8713,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((9101,6427,6751)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{11}{20}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9450 }(3223, a) \) \(1\)\(1\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{7}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9450 }(3223,a) \;\) at \(\;a = \) e.g. 2