Properties

Label 2304.1969
Modulus $2304$
Conductor $576$
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(2304, base_ring=CyclotomicField(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,39,32]))
 
pari: [g,chi] = znchar(Mod(1969,2304))
 

Basic properties

Modulus: \(2304\)
Conductor: \(576\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{576}(277,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2304.bq

\(\chi_{2304}(49,\cdot)\) \(\chi_{2304}(241,\cdot)\) \(\chi_{2304}(337,\cdot)\) \(\chi_{2304}(529,\cdot)\) \(\chi_{2304}(625,\cdot)\) \(\chi_{2304}(817,\cdot)\) \(\chi_{2304}(913,\cdot)\) \(\chi_{2304}(1105,\cdot)\) \(\chi_{2304}(1201,\cdot)\) \(\chi_{2304}(1393,\cdot)\) \(\chi_{2304}(1489,\cdot)\) \(\chi_{2304}(1681,\cdot)\) \(\chi_{2304}(1777,\cdot)\) \(\chi_{2304}(1969,\cdot)\) \(\chi_{2304}(2065,\cdot)\) \(\chi_{2304}(2257,\cdot)\)

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

\((1279,2053,1793)\) → \((1,e\left(\frac{13}{16}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 2304 }(1969, a) \) \(1\)\(1\)\(e\left(\frac{7}{48}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{35}{48}\right)\)\(e\left(\frac{25}{48}\right)\)\(-i\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{29}{48}\right)\)\(e\left(\frac{5}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2304 }(1969,a) \;\) at \(\;a = \) e.g. 2