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Properties

Label 1860.dk
Modulus $1860$
Conductor $465$
Order $60$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([0,30,45,16])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(113,1860)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(1860\)
Conductor: \(465\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(60\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from 465.bt
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(7\) \(11\) \(13\) \(17\) \(19\) \(23\) \(29\) \(37\) \(41\) \(43\)
\(\chi_{1860}(113,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{19}{60}\right)\)
\(\chi_{1860}(173,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{43}{60}\right)\)
\(\chi_{1860}(257,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{1}{60}\right)\)
\(\chi_{1860}(293,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{11}{60}\right)\)
\(\chi_{1860}(317,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{29}{60}\right)\)
\(\chi_{1860}(413,\cdot)\) \(1\) \(1\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{7}{60}\right)\)
\(\chi_{1860}(617,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{53}{60}\right)\)
\(\chi_{1860}(857,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{49}{60}\right)\)
\(\chi_{1860}(917,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{13}{60}\right)\)
\(\chi_{1860}(1037,\cdot)\) \(1\) \(1\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{41}{60}\right)\)
\(\chi_{1860}(1073,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{47}{60}\right)\)
\(\chi_{1860}(1157,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{37}{60}\right)\)
\(\chi_{1860}(1373,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{31}{60}\right)\)
\(\chi_{1860}(1433,\cdot)\) \(1\) \(1\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{59}{60}\right)\)
\(\chi_{1860}(1733,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{23}{60}\right)\)
\(\chi_{1860}(1817,\cdot)\) \(1\) \(1\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{17}{60}\right)\)