Properties

Label 2304.467
Modulus $2304$
Conductor $768$
Order $64$
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(64))
 
M = H._module
 
chi = DirichletCharacter(H, M([32,7,32]))
 
pari: [g,chi] = znchar(Mod(467,2304))
 

Basic properties

Modulus: \(2304\)
Conductor: \(768\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(64\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{768}(467,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2304.bu

\(\chi_{2304}(35,\cdot)\) \(\chi_{2304}(107,\cdot)\) \(\chi_{2304}(179,\cdot)\) \(\chi_{2304}(251,\cdot)\) \(\chi_{2304}(323,\cdot)\) \(\chi_{2304}(395,\cdot)\) \(\chi_{2304}(467,\cdot)\) \(\chi_{2304}(539,\cdot)\) \(\chi_{2304}(611,\cdot)\) \(\chi_{2304}(683,\cdot)\) \(\chi_{2304}(755,\cdot)\) \(\chi_{2304}(827,\cdot)\) \(\chi_{2304}(899,\cdot)\) \(\chi_{2304}(971,\cdot)\) \(\chi_{2304}(1043,\cdot)\) \(\chi_{2304}(1115,\cdot)\) \(\chi_{2304}(1187,\cdot)\) \(\chi_{2304}(1259,\cdot)\) \(\chi_{2304}(1331,\cdot)\) \(\chi_{2304}(1403,\cdot)\) \(\chi_{2304}(1475,\cdot)\) \(\chi_{2304}(1547,\cdot)\) \(\chi_{2304}(1619,\cdot)\) \(\chi_{2304}(1691,\cdot)\) \(\chi_{2304}(1763,\cdot)\) \(\chi_{2304}(1835,\cdot)\) \(\chi_{2304}(1907,\cdot)\) \(\chi_{2304}(1979,\cdot)\) \(\chi_{2304}(2051,\cdot)\) \(\chi_{2304}(2123,\cdot)\) ...

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: $\Q(\zeta_{64})$
Fixed field: Number field defined by a degree 64 polynomial

Values on generators

\((1279,2053,1793)\) → \((-1,e\left(\frac{7}{64}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 2304 }(467, a) \) \(1\)\(1\)\(e\left(\frac{39}{64}\right)\)\(e\left(\frac{19}{32}\right)\)\(e\left(\frac{19}{64}\right)\)\(e\left(\frac{9}{64}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{1}{64}\right)\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{7}{32}\right)\)\(e\left(\frac{61}{64}\right)\)\(e\left(\frac{3}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2304 }(467,a) \;\) at \(\;a = \) e.g. 2