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Properties

Label 1309.ck
Modulus $1309$
Conductor $1309$
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(1309, base_ring=CyclotomicField(48)) M = H._module chi = DirichletCharacter(H, M([8,24,9])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(10,1309)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(1309\)
Conductor: \(1309\)
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 \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(8\) \(9\) \(10\) \(12\) \(13\)
\(\chi_{1309}(10,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{13}{48}\right)\) \(-i\)
\(\chi_{1309}(54,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{47}{48}\right)\) \(i\)
\(\chi_{1309}(131,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{35}{48}\right)\) \(i\)
\(\chi_{1309}(164,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{47}{48}\right)\) \(e\left(\frac{25}{48}\right)\) \(-i\)
\(\chi_{1309}(241,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{31}{48}\right)\) \(i\)
\(\chi_{1309}(318,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{47}{48}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{5}{48}\right)\) \(e\left(\frac{19}{48}\right)\) \(i\)
\(\chi_{1309}(362,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{13}{48}\right)\) \(e\left(\frac{11}{48}\right)\) \(i\)
\(\chi_{1309}(439,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{19}{48}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{47}{48}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{23}{48}\right)\) \(i\)
\(\chi_{1309}(516,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{17}{48}\right)\) \(-i\)
\(\chi_{1309}(549,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{19}{48}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{43}{48}\right)\) \(i\)
\(\chi_{1309}(626,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{7}{48}\right)\) \(i\)
\(\chi_{1309}(670,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{19}{48}\right)\) \(e\left(\frac{5}{48}\right)\) \(-i\)
\(\chi_{1309}(703,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{5}{48}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{1}{48}\right)\) \(-i\)
\(\chi_{1309}(857,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{13}{48}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{37}{48}\right)\) \(-i\)
\(\chi_{1309}(1132,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{48}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{29}{48}\right)\) \(-i\)
\(\chi_{1309}(1286,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{13}{48}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{41}{48}\right)\) \(-i\)