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Properties

Label 325.bm
Modulus $325$
Conductor $325$
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(325, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([24,25])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(6,325)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(325\)
Conductor: \(325\)
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 \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(7\) \(8\) \(9\) \(11\) \(12\) \(14\)
\(\chi_{325}(6,\cdot)\) \(-1\) \(1\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{2}{5}\right)\)
\(\chi_{325}(11,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{4}{5}\right)\)
\(\chi_{325}(41,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{5}\right)\)
\(\chi_{325}(46,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{5}\right)\)
\(\chi_{325}(71,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{5}\right)\)
\(\chi_{325}(106,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{2}{5}\right)\)
\(\chi_{325}(111,\cdot)\) \(-1\) \(1\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{4}{5}\right)\)
\(\chi_{325}(136,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{4}{5}\right)\)
\(\chi_{325}(141,\cdot)\) \(-1\) \(1\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{5}\right)\)
\(\chi_{325}(171,\cdot)\) \(-1\) \(1\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{5}\right)\)
\(\chi_{325}(206,\cdot)\) \(-1\) \(1\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{2}{5}\right)\)
\(\chi_{325}(236,\cdot)\) \(-1\) \(1\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{4}{5}\right)\)
\(\chi_{325}(241,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{5}\right)\)
\(\chi_{325}(266,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{5}\right)\)
\(\chi_{325}(271,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{5}\right)\)
\(\chi_{325}(306,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{2}{5}\right)\)