Properties

Label 1224.607
Modulus $1224$
Conductor $612$
Order $48$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1224, base_ring=CyclotomicField(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([24,0,16,39]))
 
pari: [g,chi] = znchar(Mod(607,1224))
 

Basic properties

Modulus: \(1224\)
Conductor: \(612\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{612}(607,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1224.cv

\(\chi_{1224}(7,\cdot)\) \(\chi_{1224}(31,\cdot)\) \(\chi_{1224}(79,\cdot)\) \(\chi_{1224}(175,\cdot)\) \(\chi_{1224}(295,\cdot)\) \(\chi_{1224}(367,\cdot)\) \(\chi_{1224}(439,\cdot)\) \(\chi_{1224}(583,\cdot)\) \(\chi_{1224}(607,\cdot)\) \(\chi_{1224}(751,\cdot)\) \(\chi_{1224}(823,\cdot)\) \(\chi_{1224}(895,\cdot)\) \(\chi_{1224}(1015,\cdot)\) \(\chi_{1224}(1111,\cdot)\) \(\chi_{1224}(1159,\cdot)\) \(\chi_{1224}(1183,\cdot)\)

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: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((919,613,137,649)\) → \((-1,1,e\left(\frac{1}{3}\right),e\left(\frac{13}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 1224 }(607, a) \) \(1\)\(1\)\(e\left(\frac{35}{48}\right)\)\(e\left(\frac{37}{48}\right)\)\(e\left(\frac{25}{48}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{17}{48}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{43}{48}\right)\)\(e\left(\frac{23}{48}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1224 }(607,a) \;\) at \(\;a = \) e.g. 2