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Properties

Label 8400.kz
Modulus $8400$
Conductor $8400$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8400, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([30,45,30,57,10])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(563,8400)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(8400\)
Conductor: \(8400\)
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: 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 \(-1\) \(1\) \(11\) \(13\) \(17\) \(19\) \(23\) \(29\) \(31\) \(37\) \(41\) \(43\)
\(\chi_{8400}(563,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{3}{10}\right)\) \(-1\)
\(\chi_{8400}(803,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{9}{10}\right)\) \(-1\)
\(\chi_{8400}(1067,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{1}{10}\right)\) \(-1\)
\(\chi_{8400}(2483,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{1}{10}\right)\) \(-1\)
\(\chi_{8400}(2747,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{9}{10}\right)\) \(-1\)
\(\chi_{8400}(2987,\cdot)\) \(1\) \(1\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{3}{10}\right)\) \(-1\)
\(\chi_{8400}(3923,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{7}{10}\right)\) \(-1\)
\(\chi_{8400}(4163,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{3}{10}\right)\) \(-1\)
\(\chi_{8400}(4427,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{7}{10}\right)\) \(-1\)
\(\chi_{8400}(4667,\cdot)\) \(1\) \(1\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{1}{10}\right)\) \(-1\)
\(\chi_{8400}(5603,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{9}{10}\right)\) \(-1\)
\(\chi_{8400}(6347,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{9}{10}\right)\) \(-1\)
\(\chi_{8400}(7283,\cdot)\) \(1\) \(1\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{1}{10}\right)\) \(-1\)
\(\chi_{8400}(7523,\cdot)\) \(1\) \(1\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{7}{10}\right)\) \(-1\)
\(\chi_{8400}(7787,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{3}{10}\right)\) \(-1\)
\(\chi_{8400}(8027,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{7}{10}\right)\) \(-1\)