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Properties

Label 1575.en
Modulus $1575$
Conductor $1575$
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(1575, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([10,57,10])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(38,1575)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(1575\)
Conductor: \(1575\)
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 \(-1\) \(1\) \(2\) \(4\) \(8\) \(11\) \(13\) \(16\) \(17\) \(19\) \(22\) \(23\)
\(\chi_{1575}(38,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{37}{60}\right)\)
\(\chi_{1575}(227,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{43}{60}\right)\)
\(\chi_{1575}(353,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{1}{60}\right)\)
\(\chi_{1575}(383,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{29}{60}\right)\)
\(\chi_{1575}(542,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{19}{60}\right)\)
\(\chi_{1575}(572,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{11}{60}\right)\)
\(\chi_{1575}(698,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{53}{60}\right)\)
\(\chi_{1575}(887,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{47}{60}\right)\)
\(\chi_{1575}(983,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{49}{60}\right)\)
\(\chi_{1575}(1013,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{17}{60}\right)\)
\(\chi_{1575}(1172,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{31}{60}\right)\)
\(\chi_{1575}(1202,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{23}{60}\right)\)
\(\chi_{1575}(1298,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{13}{60}\right)\)
\(\chi_{1575}(1328,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{41}{60}\right)\)
\(\chi_{1575}(1487,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{7}{60}\right)\)
\(\chi_{1575}(1517,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{59}{60}\right)\)