Properties

Label 2304.89
Modulus $2304$
Conductor $384$
Order $32$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(32))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,25,16]))
 
pari: [g,chi] = znchar(Mod(89,2304))
 

Basic properties

Modulus: \(2304\)
Conductor: \(384\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{384}(293,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2304.bn

\(\chi_{2304}(89,\cdot)\) \(\chi_{2304}(233,\cdot)\) \(\chi_{2304}(377,\cdot)\) \(\chi_{2304}(521,\cdot)\) \(\chi_{2304}(665,\cdot)\) \(\chi_{2304}(809,\cdot)\) \(\chi_{2304}(953,\cdot)\) \(\chi_{2304}(1097,\cdot)\) \(\chi_{2304}(1241,\cdot)\) \(\chi_{2304}(1385,\cdot)\) \(\chi_{2304}(1529,\cdot)\) \(\chi_{2304}(1673,\cdot)\) \(\chi_{2304}(1817,\cdot)\) \(\chi_{2304}(1961,\cdot)\) \(\chi_{2304}(2105,\cdot)\) \(\chi_{2304}(2249,\cdot)\)

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: \(\Q(\zeta_{32})\)
Fixed field: 32.0.135104323545903136978453058557785670637514001130337144105502507008.1

Values on generators

\((1279,2053,1793)\) → \((1,e\left(\frac{25}{32}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 2304 }(89, a) \) \(-1\)\(1\)\(e\left(\frac{9}{32}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{29}{32}\right)\)\(e\left(\frac{23}{32}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{31}{32}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{19}{32}\right)\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2304 }(89,a) \;\) at \(\;a = \) e.g. 2