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Properties

Label 221.bj
Modulus $221$
Conductor $221$
Order $48$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(221, base_ring=CyclotomicField(48)) M = H._module chi = DirichletCharacter(H, M([16,3])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(3,221)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(221\)
Conductor: \(221\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(48\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content 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 \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{221}(3,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{19}{48}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{37}{48}\right)\)
\(\chi_{221}(22,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{47}{48}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{41}{48}\right)\)
\(\chi_{221}(29,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{1}{48}\right)\)
\(\chi_{221}(48,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{29}{48}\right)\)
\(\chi_{221}(61,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{19}{48}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{47}{48}\right)\)
\(\chi_{221}(74,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{19}{48}\right)\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{11}{48}\right)\)
\(\chi_{221}(107,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{5}{48}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{13}{48}\right)\) \(e\left(\frac{25}{48}\right)\)
\(\chi_{221}(113,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{5}{48}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{47}{48}\right)\) \(e\left(\frac{35}{48}\right)\)
\(\chi_{221}(126,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{23}{48}\right)\)
\(\chi_{221}(133,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{5}{48}\right)\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{13}{48}\right)\)
\(\chi_{221}(139,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{13}{48}\right)\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{5}{48}\right)\)
\(\chi_{221}(146,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{31}{48}\right)\)
\(\chi_{221}(159,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{13}{48}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{47}{48}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{43}{48}\right)\)
\(\chi_{221}(165,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{13}{48}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{5}{48}\right)\) \(e\left(\frac{17}{48}\right)\)
\(\chi_{221}(198,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{19}{48}\right)\)
\(\chi_{221}(211,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{47}{48}\right)\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{19}{48}\right)\) \(e\left(\frac{7}{48}\right)\)