Properties

Label 1375.bx
Modulus $1375$
Conductor $1375$
Order $50$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1375, base_ring=CyclotomicField(50))
 
M = H._module
 
chi = DirichletCharacter(H, M([23,45]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(39,1375))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(1375\)
Conductor: \(1375\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(50\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{25})\)
Fixed field: Number field defined by a degree 50 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(7\) \(8\) \(9\) \(12\) \(13\) \(14\)
\(\chi_{1375}(39,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{19}{25}\right)\)
\(\chi_{1375}(129,\cdot)\) \(-1\) \(1\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{3}{25}\right)\)
\(\chi_{1375}(134,\cdot)\) \(-1\) \(1\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{21}{25}\right)\)
\(\chi_{1375}(194,\cdot)\) \(-1\) \(1\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{7}{25}\right)\)
\(\chi_{1375}(314,\cdot)\) \(-1\) \(1\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{9}{25}\right)\)
\(\chi_{1375}(404,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{18}{25}\right)\)
\(\chi_{1375}(409,\cdot)\) \(-1\) \(1\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{11}{25}\right)\)
\(\chi_{1375}(469,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{22}{25}\right)\)
\(\chi_{1375}(589,\cdot)\) \(-1\) \(1\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{24}{25}\right)\)
\(\chi_{1375}(679,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{8}{25}\right)\)
\(\chi_{1375}(684,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{1}{25}\right)\)
\(\chi_{1375}(744,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{12}{25}\right)\)
\(\chi_{1375}(864,\cdot)\) \(-1\) \(1\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{14}{25}\right)\)
\(\chi_{1375}(954,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{23}{25}\right)\)
\(\chi_{1375}(959,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{16}{25}\right)\)
\(\chi_{1375}(1019,\cdot)\) \(-1\) \(1\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{2}{25}\right)\)
\(\chi_{1375}(1139,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{4}{25}\right)\)
\(\chi_{1375}(1229,\cdot)\) \(-1\) \(1\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{13}{25}\right)\)
\(\chi_{1375}(1234,\cdot)\) \(-1\) \(1\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{6}{25}\right)\)
\(\chi_{1375}(1294,\cdot)\) \(-1\) \(1\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{17}{25}\right)\)