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Properties

Label 475.bd
Modulus $475$
Conductor $475$
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(475, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([9,20])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(83,475)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(475\)
Conductor: \(475\)
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 \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(7\) \(8\) \(9\) \(11\) \(12\) \(13\)
\(\chi_{475}(83,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{13}{15}\right)\) \(-i\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{31}{60}\right)\)
\(\chi_{475}(87,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{14}{15}\right)\) \(i\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{53}{60}\right)\)
\(\chi_{475}(102,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{1}{15}\right)\) \(i\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{37}{60}\right)\)
\(\chi_{475}(163,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{14}{15}\right)\) \(-i\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{23}{60}\right)\)
\(\chi_{475}(178,\cdot)\) \(-1\) \(1\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{7}{15}\right)\) \(-i\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{19}{60}\right)\)
\(\chi_{475}(197,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{7}{15}\right)\) \(i\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{49}{60}\right)\)
\(\chi_{475}(258,\cdot)\) \(-1\) \(1\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{8}{15}\right)\) \(-i\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{11}{60}\right)\)
\(\chi_{475}(273,\cdot)\) \(-1\) \(1\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{1}{15}\right)\) \(-i\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{7}{60}\right)\)
\(\chi_{475}(277,\cdot)\) \(-1\) \(1\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{11}{15}\right)\) \(i\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{17}{60}\right)\)
\(\chi_{475}(292,\cdot)\) \(-1\) \(1\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{13}{15}\right)\) \(i\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{1}{60}\right)\)
\(\chi_{475}(353,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{2}{15}\right)\) \(-i\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{59}{60}\right)\)
\(\chi_{475}(372,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{2}{15}\right)\) \(i\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{29}{60}\right)\)
\(\chi_{475}(387,\cdot)\) \(-1\) \(1\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{4}{15}\right)\) \(i\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{13}{60}\right)\)
\(\chi_{475}(448,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{11}{15}\right)\) \(-i\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{47}{60}\right)\)
\(\chi_{475}(463,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{4}{15}\right)\) \(-i\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{43}{60}\right)\)
\(\chi_{475}(467,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{8}{15}\right)\) \(i\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{41}{60}\right)\)