Properties

Label 9450.7193
Modulus $9450$
Conductor $945$
Order $36$
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(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([26,27,24]))
 
pari: [g,chi] = znchar(Mod(7193,9450))
 

Basic properties

Modulus: \(9450\)
Conductor: \(945\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{945}(578,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9450.ez

\(\chi_{9450}(893,\cdot)\) \(\chi_{9450}(1157,\cdot)\) \(\chi_{9450}(2543,\cdot)\) \(\chi_{9450}(2657,\cdot)\) \(\chi_{9450}(4043,\cdot)\) \(\chi_{9450}(4307,\cdot)\) \(\chi_{9450}(5693,\cdot)\) \(\chi_{9450}(5807,\cdot)\) \(\chi_{9450}(7193,\cdot)\) \(\chi_{9450}(7457,\cdot)\) \(\chi_{9450}(8843,\cdot)\) \(\chi_{9450}(8957,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: 36.36.12458220182500526775836714711907142658919420093068625931818090744316577911376953125.2

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9450 }(7193, a) \) \(1\)\(1\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{1}{36}\right)\)\(i\)\(-1\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{5}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9450 }(7193,a) \;\) at \(\;a = \) e.g. 2