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Properties

Label 768.x
Modulus $768$
Conductor $384$
Order $32$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(32)) M = H._module chi = DirichletCharacter(H, M([0,31,16])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(41,768)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(768\)
Conductor: \(384\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(32\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from 384.x
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Related number fields

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

Characters in Galois orbit

Character \(-1\) \(1\) \(5\) \(7\) \(11\) \(13\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\)
\(\chi_{768}(41,\cdot)\) \(-1\) \(1\) \(e\left(\frac{15}{32}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{27}{32}\right)\) \(e\left(\frac{17}{32}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{9}{32}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{21}{32}\right)\) \(-i\)
\(\chi_{768}(89,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{32}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{29}{32}\right)\) \(e\left(\frac{23}{32}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{31}{32}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{19}{32}\right)\) \(i\)
\(\chi_{768}(137,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{32}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{31}{32}\right)\) \(e\left(\frac{29}{32}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{21}{32}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{17}{32}\right)\) \(-i\)
\(\chi_{768}(185,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{32}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{1}{32}\right)\) \(e\left(\frac{3}{32}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{11}{32}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{15}{32}\right)\) \(i\)
\(\chi_{768}(233,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{32}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{3}{32}\right)\) \(e\left(\frac{9}{32}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{1}{32}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{13}{32}\right)\) \(-i\)
\(\chi_{768}(281,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{32}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{5}{32}\right)\) \(e\left(\frac{15}{32}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{23}{32}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{11}{32}\right)\) \(i\)
\(\chi_{768}(329,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{32}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{7}{32}\right)\) \(e\left(\frac{21}{32}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{13}{32}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{9}{32}\right)\) \(-i\)
\(\chi_{768}(377,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{32}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{9}{32}\right)\) \(e\left(\frac{27}{32}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{3}{32}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{7}{32}\right)\) \(i\)
\(\chi_{768}(425,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{32}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{11}{32}\right)\) \(e\left(\frac{1}{32}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{25}{32}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{5}{32}\right)\) \(-i\)
\(\chi_{768}(473,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{32}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{13}{32}\right)\) \(e\left(\frac{7}{32}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{15}{32}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{3}{32}\right)\) \(i\)
\(\chi_{768}(521,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{32}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{15}{32}\right)\) \(e\left(\frac{13}{32}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{5}{32}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{1}{32}\right)\) \(-i\)
\(\chi_{768}(569,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{32}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{17}{32}\right)\) \(e\left(\frac{19}{32}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{27}{32}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{31}{32}\right)\) \(i\)
\(\chi_{768}(617,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{32}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{19}{32}\right)\) \(e\left(\frac{25}{32}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{17}{32}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{29}{32}\right)\) \(-i\)
\(\chi_{768}(665,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{32}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{21}{32}\right)\) \(e\left(\frac{31}{32}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{7}{32}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{27}{32}\right)\) \(i\)
\(\chi_{768}(713,\cdot)\) \(-1\) \(1\) \(e\left(\frac{27}{32}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{23}{32}\right)\) \(e\left(\frac{5}{32}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{29}{32}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{25}{32}\right)\) \(-i\)
\(\chi_{768}(761,\cdot)\) \(-1\) \(1\) \(e\left(\frac{21}{32}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{25}{32}\right)\) \(e\left(\frac{11}{32}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{19}{32}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{23}{32}\right)\) \(i\)