Properties

Label 384.x
Modulus $384$
Conductor $384$
Order $32$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(384, base_ring=CyclotomicField(32))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,1,16]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(5,384))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(384\)
Conductor: \(384\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

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

Characters in Galois orbit

Character \(-1\) \(1\) \(5\) \(7\) \(11\) \(13\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\)
\(\chi_{384}(5,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{32}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{5}{32}\right)\) \(e\left(\frac{15}{32}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{23}{32}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{11}{32}\right)\) \(i\)
\(\chi_{384}(29,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{32}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{7}{32}\right)\) \(e\left(\frac{21}{32}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{13}{32}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{9}{32}\right)\) \(-i\)
\(\chi_{384}(53,\cdot)\) \(-1\) \(1\) \(e\left(\frac{21}{32}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{25}{32}\right)\) \(e\left(\frac{11}{32}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{19}{32}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{23}{32}\right)\) \(i\)
\(\chi_{384}(77,\cdot)\) \(-1\) \(1\) \(e\left(\frac{15}{32}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{27}{32}\right)\) \(e\left(\frac{17}{32}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{9}{32}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{21}{32}\right)\) \(-i\)
\(\chi_{384}(101,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{32}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{13}{32}\right)\) \(e\left(\frac{7}{32}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{15}{32}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{3}{32}\right)\) \(i\)
\(\chi_{384}(125,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{32}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{15}{32}\right)\) \(e\left(\frac{13}{32}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{5}{32}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{1}{32}\right)\) \(-i\)
\(\chi_{384}(149,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{32}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{1}{32}\right)\) \(e\left(\frac{3}{32}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{11}{32}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{15}{32}\right)\) \(i\)
\(\chi_{384}(173,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{32}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{3}{32}\right)\) \(e\left(\frac{9}{32}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{1}{32}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{13}{32}\right)\) \(-i\)
\(\chi_{384}(197,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{32}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{21}{32}\right)\) \(e\left(\frac{31}{32}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{7}{32}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{27}{32}\right)\) \(i\)
\(\chi_{384}(221,\cdot)\) \(-1\) \(1\) \(e\left(\frac{27}{32}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{23}{32}\right)\) \(e\left(\frac{5}{32}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{29}{32}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{25}{32}\right)\) \(-i\)
\(\chi_{384}(245,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{32}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{9}{32}\right)\) \(e\left(\frac{27}{32}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{3}{32}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{7}{32}\right)\) \(i\)
\(\chi_{384}(269,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{32}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{11}{32}\right)\) \(e\left(\frac{1}{32}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{25}{32}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{5}{32}\right)\) \(-i\)
\(\chi_{384}(293,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{32}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{29}{32}\right)\) \(e\left(\frac{23}{32}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{31}{32}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{19}{32}\right)\) \(i\)
\(\chi_{384}(317,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{32}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{31}{32}\right)\) \(e\left(\frac{29}{32}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{21}{32}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{17}{32}\right)\) \(-i\)
\(\chi_{384}(341,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{32}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{17}{32}\right)\) \(e\left(\frac{19}{32}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{27}{32}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{31}{32}\right)\) \(i\)
\(\chi_{384}(365,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{32}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{19}{32}\right)\) \(e\left(\frac{25}{32}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{17}{32}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{29}{32}\right)\) \(-i\)