Properties

Label 1175.24
Modulus $1175$
Conductor $235$
Order $46$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1175, base_ring=CyclotomicField(46))
 
M = H._module
 
chi = DirichletCharacter(H, M([23,28]))
 
pari: [g,chi] = znchar(Mod(24,1175))
 

Basic properties

Modulus: \(1175\)
Conductor: \(235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(46\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{235}(24,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1175.o

\(\chi_{1175}(24,\cdot)\) \(\chi_{1175}(49,\cdot)\) \(\chi_{1175}(74,\cdot)\) \(\chi_{1175}(149,\cdot)\) \(\chi_{1175}(224,\cdot)\) \(\chi_{1175}(249,\cdot)\) \(\chi_{1175}(299,\cdot)\) \(\chi_{1175}(324,\cdot)\) \(\chi_{1175}(474,\cdot)\) \(\chi_{1175}(524,\cdot)\) \(\chi_{1175}(549,\cdot)\) \(\chi_{1175}(674,\cdot)\) \(\chi_{1175}(824,\cdot)\) \(\chi_{1175}(849,\cdot)\) \(\chi_{1175}(874,\cdot)\) \(\chi_{1175}(899,\cdot)\) \(\chi_{1175}(949,\cdot)\) \(\chi_{1175}(974,\cdot)\) \(\chi_{1175}(999,\cdot)\) \(\chi_{1175}(1024,\cdot)\) \(\chi_{1175}(1099,\cdot)\) \(\chi_{1175}(1149,\cdot)\)

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: \(\Q(\zeta_{23})\)
Fixed field: 46.46.445262221645814097378614331350194306504897709623466822770698795048558574688434600830078125.1

Values on generators

\((377,851)\) → \((-1,e\left(\frac{14}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1175 }(24, a) \) \(1\)\(1\)\(e\left(\frac{21}{46}\right)\)\(e\left(\frac{31}{46}\right)\)\(e\left(\frac{21}{23}\right)\)\(e\left(\frac{3}{23}\right)\)\(e\left(\frac{45}{46}\right)\)\(e\left(\frac{17}{46}\right)\)\(e\left(\frac{8}{23}\right)\)\(e\left(\frac{6}{23}\right)\)\(e\left(\frac{27}{46}\right)\)\(e\left(\frac{9}{46}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1175 }(24,a) \;\) at \(\;a = \) e.g. 2