Properties

Label 9450.3271
Modulus $9450$
Conductor $4725$
Order $45$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9450, base_ring=CyclotomicField(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,54,30]))
 
pari: [g,chi] = znchar(Mod(3271,9450))
 

Basic properties

Modulus: \(9450\)
Conductor: \(4725\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(45\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4725}(3271,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9450.fk

\(\chi_{9450}(121,\cdot)\) \(\chi_{9450}(781,\cdot)\) \(\chi_{9450}(1381,\cdot)\) \(\chi_{9450}(1411,\cdot)\) \(\chi_{9450}(2011,\cdot)\) \(\chi_{9450}(2041,\cdot)\) \(\chi_{9450}(2641,\cdot)\) \(\chi_{9450}(2671,\cdot)\) \(\chi_{9450}(3271,\cdot)\) \(\chi_{9450}(3931,\cdot)\) \(\chi_{9450}(4531,\cdot)\) \(\chi_{9450}(4561,\cdot)\) \(\chi_{9450}(5161,\cdot)\) \(\chi_{9450}(5191,\cdot)\) \(\chi_{9450}(5791,\cdot)\) \(\chi_{9450}(5821,\cdot)\) \(\chi_{9450}(6421,\cdot)\) \(\chi_{9450}(7081,\cdot)\) \(\chi_{9450}(7681,\cdot)\) \(\chi_{9450}(7711,\cdot)\) \(\chi_{9450}(8311,\cdot)\) \(\chi_{9450}(8341,\cdot)\) \(\chi_{9450}(8941,\cdot)\) \(\chi_{9450}(8971,\cdot)\)

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 45 polynomial

Values on generators

\((9101,6427,6751)\) → \((e\left(\frac{1}{9}\right),e\left(\frac{3}{5}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9450 }(3271, a) \) \(1\)\(1\)\(e\left(\frac{17}{45}\right)\)\(e\left(\frac{13}{45}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{22}{45}\right)\)\(e\left(\frac{14}{45}\right)\)\(e\left(\frac{16}{45}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{13}{45}\right)\)\(e\left(\frac{4}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9450 }(3271,a) \;\) at \(\;a = \) e.g. 2