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Properties

Label 1800.dm
Modulus $1800$
Conductor $1800$
Order $60$
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(1800, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([30,30,10,9])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(83,1800)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(1800\)
Conductor: \(1800\)
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: 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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(7\) \(11\) \(13\) \(17\) \(19\) \(23\) \(29\) \(31\) \(37\) \(41\)
\(\chi_{1800}(83,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{13}{30}\right)\)
\(\chi_{1800}(203,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{17}{30}\right)\)
\(\chi_{1800}(227,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{1}{30}\right)\)
\(\chi_{1800}(347,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{17}{30}\right)\)
\(\chi_{1800}(563,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{29}{30}\right)\)
\(\chi_{1800}(587,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{19}{30}\right)\)
\(\chi_{1800}(803,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{7}{30}\right)\)
\(\chi_{1800}(923,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{11}{30}\right)\)
\(\chi_{1800}(947,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{7}{30}\right)\)
\(\chi_{1800}(1067,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{23}{30}\right)\)
\(\chi_{1800}(1163,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{19}{30}\right)\)
\(\chi_{1800}(1283,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{23}{30}\right)\)
\(\chi_{1800}(1427,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{11}{30}\right)\)
\(\chi_{1800}(1523,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{1}{30}\right)\)
\(\chi_{1800}(1667,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{13}{30}\right)\)
\(\chi_{1800}(1787,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{29}{30}\right)\)