Properties

Label 1728.bz
Modulus $1728$
Conductor $576$
Order $48$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1728, base_ring=CyclotomicField(48))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,9,40]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(125,1728))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(1728\)
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 576.bn
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(5\) \(7\) \(11\) \(13\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\)
\(\chi_{1728}(125,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{23}{48}\right)\) \(-i\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{1728}(197,\cdot)\) \(-1\) \(1\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{13}{48}\right)\) \(i\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{1728}(341,\cdot)\) \(-1\) \(1\) \(e\left(\frac{47}{48}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{41}{48}\right)\) \(i\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{1728}(413,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{31}{48}\right)\) \(-i\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{1728}(557,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{11}{48}\right)\) \(-i\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{1728}(629,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{1}{48}\right)\) \(i\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{1728}(773,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{29}{48}\right)\) \(i\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{1728}(845,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{19}{48}\right)\) \(-i\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{1728}(989,\cdot)\) \(-1\) \(1\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{13}{48}\right)\) \(e\left(\frac{47}{48}\right)\) \(-i\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{19}{48}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{1728}(1061,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{48}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{47}{48}\right)\) \(e\left(\frac{37}{48}\right)\) \(i\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{1728}(1205,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{19}{48}\right)\) \(e\left(\frac{17}{48}\right)\) \(i\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{13}{48}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{1728}(1277,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{5}{48}\right)\) \(e\left(\frac{7}{48}\right)\) \(-i\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{1728}(1421,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{48}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{35}{48}\right)\) \(-i\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{1728}(1493,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{25}{48}\right)\) \(i\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{5}{48}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{1728}(1637,\cdot)\) \(-1\) \(1\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{5}{48}\right)\) \(i\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{1728}(1709,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{48}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{43}{48}\right)\) \(-i\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{47}{48}\right)\) \(e\left(\frac{5}{6}\right)\)