Properties

Label 9450.1429
Modulus $9450$
Conductor $675$
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([50,9,0]))
 
pari: [g,chi] = znchar(Mod(1429,9450))
 

Basic properties

Modulus: \(9450\)
Conductor: \(675\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{675}(79,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9450.gl

\(\chi_{9450}(169,\cdot)\) \(\chi_{9450}(589,\cdot)\) \(\chi_{9450}(1219,\cdot)\) \(\chi_{9450}(1429,\cdot)\) \(\chi_{9450}(2059,\cdot)\) \(\chi_{9450}(2479,\cdot)\) \(\chi_{9450}(2689,\cdot)\) \(\chi_{9450}(3109,\cdot)\) \(\chi_{9450}(3319,\cdot)\) \(\chi_{9450}(3739,\cdot)\) \(\chi_{9450}(4369,\cdot)\) \(\chi_{9450}(4579,\cdot)\) \(\chi_{9450}(5209,\cdot)\) \(\chi_{9450}(5629,\cdot)\) \(\chi_{9450}(5839,\cdot)\) \(\chi_{9450}(6259,\cdot)\) \(\chi_{9450}(6469,\cdot)\) \(\chi_{9450}(6889,\cdot)\) \(\chi_{9450}(7519,\cdot)\) \(\chi_{9450}(7729,\cdot)\) \(\chi_{9450}(8359,\cdot)\) \(\chi_{9450}(8779,\cdot)\) \(\chi_{9450}(8989,\cdot)\) \(\chi_{9450}(9409,\cdot)\)

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

\((9101,6427,6751)\) → \((e\left(\frac{5}{9}\right),e\left(\frac{1}{10}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9450 }(1429, a) \) \(1\)\(1\)\(e\left(\frac{37}{45}\right)\)\(e\left(\frac{31}{90}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{19}{90}\right)\)\(e\left(\frac{34}{45}\right)\)\(e\left(\frac{41}{45}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{38}{45}\right)\)\(e\left(\frac{13}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9450 }(1429,a) \;\) at \(\;a = \) e.g. 2