Properties

Label 2240.2231
Modulus $2240$
Conductor $224$
Order $24$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([12,9,0,20]))
 
pari: [g,chi] = znchar(Mod(2231,2240))
 

Basic properties

Modulus: \(2240\)
Conductor: \(224\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{224}(131,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2240.ej

\(\chi_{2240}(311,\cdot)\) \(\chi_{2240}(551,\cdot)\) \(\chi_{2240}(871,\cdot)\) \(\chi_{2240}(1111,\cdot)\) \(\chi_{2240}(1431,\cdot)\) \(\chi_{2240}(1671,\cdot)\) \(\chi_{2240}(1991,\cdot)\) \(\chi_{2240}(2231,\cdot)\)

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: \(\Q(\zeta_{24})\)
Fixed field: 24.24.790224330201082600125157415256880139617697792.1

Values on generators

\((1471,1541,897,1921)\) → \((-1,e\left(\frac{3}{8}\right),1,e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 2240 }(2231, a) \) \(1\)\(1\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2240 }(2231,a) \;\) at \(\;a = \) e.g. 2