Properties

Label 9450.11
Modulus $9450$
Conductor $4725$
Order $90$
Real no
Primitive no
Minimal yes
Parity odd

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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([65,72,60]))
 
pari: [g,chi] = znchar(Mod(11,9450))
 

Basic properties

Modulus: \(9450\)
Conductor: \(4725\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4725}(11,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9450.gv

\(\chi_{9450}(11,\cdot)\) \(\chi_{9450}(641,\cdot)\) \(\chi_{9450}(1031,\cdot)\) \(\chi_{9450}(1271,\cdot)\) \(\chi_{9450}(1661,\cdot)\) \(\chi_{9450}(2291,\cdot)\) \(\chi_{9450}(2531,\cdot)\) \(\chi_{9450}(2921,\cdot)\) \(\chi_{9450}(3161,\cdot)\) \(\chi_{9450}(3791,\cdot)\) \(\chi_{9450}(4181,\cdot)\) \(\chi_{9450}(4421,\cdot)\) \(\chi_{9450}(4811,\cdot)\) \(\chi_{9450}(5441,\cdot)\) \(\chi_{9450}(5681,\cdot)\) \(\chi_{9450}(6071,\cdot)\) \(\chi_{9450}(6311,\cdot)\) \(\chi_{9450}(6941,\cdot)\) \(\chi_{9450}(7331,\cdot)\) \(\chi_{9450}(7571,\cdot)\) \(\chi_{9450}(7961,\cdot)\) \(\chi_{9450}(8591,\cdot)\) \(\chi_{9450}(8831,\cdot)\) \(\chi_{9450}(9221,\cdot)\)

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

\((9101,6427,6751)\) → \((e\left(\frac{13}{18}\right),e\left(\frac{4}{5}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9450 }(11, a) \) \(-1\)\(1\)\(e\left(\frac{77}{90}\right)\)\(e\left(\frac{44}{45}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{7}{90}\right)\)\(e\left(\frac{29}{90}\right)\)\(e\left(\frac{23}{45}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{43}{90}\right)\)\(e\left(\frac{8}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9450 }(11,a) \;\) at \(\;a = \) e.g. 2