Properties

Label 473.be
Modulus $473$
Conductor $473$
Order $210$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(473, base_ring=CyclotomicField(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([168,5]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(3,473))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(473\)
Conductor: \(473\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(210\)
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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 210 polynomial (not computed)

First 31 of 48 characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(12\)
\(\chi_{473}(3,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{70}\right)\) \(e\left(\frac{89}{210}\right)\) \(e\left(\frac{31}{35}\right)\) \(e\left(\frac{167}{210}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{23}{70}\right)\) \(e\left(\frac{89}{105}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{13}{42}\right)\)
\(\chi_{473}(5,\cdot)\) \(-1\) \(1\) \(e\left(\frac{33}{70}\right)\) \(e\left(\frac{167}{210}\right)\) \(e\left(\frac{33}{35}\right)\) \(e\left(\frac{101}{210}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{29}{70}\right)\) \(e\left(\frac{62}{105}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{31}{42}\right)\)
\(\chi_{473}(20,\cdot)\) \(-1\) \(1\) \(e\left(\frac{27}{70}\right)\) \(e\left(\frac{143}{210}\right)\) \(e\left(\frac{27}{35}\right)\) \(e\left(\frac{89}{210}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{11}{70}\right)\) \(e\left(\frac{38}{105}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{19}{42}\right)\)
\(\chi_{473}(26,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{70}\right)\) \(e\left(\frac{1}{210}\right)\) \(e\left(\frac{9}{35}\right)\) \(e\left(\frac{193}{210}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{27}{70}\right)\) \(e\left(\frac{1}{105}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{11}{42}\right)\)
\(\chi_{473}(48,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{70}\right)\) \(e\left(\frac{41}{210}\right)\) \(e\left(\frac{19}{35}\right)\) \(e\left(\frac{143}{210}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{57}{70}\right)\) \(e\left(\frac{41}{105}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{31}{42}\right)\)
\(\chi_{473}(69,\cdot)\) \(-1\) \(1\) \(e\left(\frac{51}{70}\right)\) \(e\left(\frac{169}{210}\right)\) \(e\left(\frac{16}{35}\right)\) \(e\left(\frac{67}{210}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{13}{70}\right)\) \(e\left(\frac{64}{105}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{11}{42}\right)\)
\(\chi_{473}(71,\cdot)\) \(-1\) \(1\) \(e\left(\frac{43}{70}\right)\) \(e\left(\frac{67}{210}\right)\) \(e\left(\frac{8}{35}\right)\) \(e\left(\frac{121}{210}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{59}{70}\right)\) \(e\left(\frac{67}{105}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{23}{42}\right)\)
\(\chi_{473}(91,\cdot)\) \(-1\) \(1\) \(e\left(\frac{61}{70}\right)\) \(e\left(\frac{209}{210}\right)\) \(e\left(\frac{26}{35}\right)\) \(e\left(\frac{17}{210}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{43}{70}\right)\) \(e\left(\frac{104}{105}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{31}{42}\right)\)
\(\chi_{473}(104,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{70}\right)\) \(e\left(\frac{187}{210}\right)\) \(e\left(\frac{3}{35}\right)\) \(e\left(\frac{181}{210}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{9}{70}\right)\) \(e\left(\frac{82}{105}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{41}{42}\right)\)
\(\chi_{473}(114,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{70}\right)\) \(e\left(\frac{151}{210}\right)\) \(e\left(\frac{29}{35}\right)\) \(e\left(\frac{163}{210}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{17}{70}\right)\) \(e\left(\frac{46}{105}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{23}{42}\right)\)
\(\chi_{473}(115,\cdot)\) \(-1\) \(1\) \(e\left(\frac{53}{70}\right)\) \(e\left(\frac{37}{210}\right)\) \(e\left(\frac{18}{35}\right)\) \(e\left(\frac{1}{210}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{19}{70}\right)\) \(e\left(\frac{37}{105}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{29}{42}\right)\)
\(\chi_{473}(119,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{70}\right)\) \(e\left(\frac{113}{210}\right)\) \(e\left(\frac{2}{35}\right)\) \(e\left(\frac{179}{210}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{41}{70}\right)\) \(e\left(\frac{8}{105}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{25}{42}\right)\)
\(\chi_{473}(141,\cdot)\) \(-1\) \(1\) \(e\left(\frac{67}{70}\right)\) \(e\left(\frac{23}{210}\right)\) \(e\left(\frac{32}{35}\right)\) \(e\left(\frac{29}{210}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{61}{70}\right)\) \(e\left(\frac{23}{105}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{1}{42}\right)\)
\(\chi_{473}(147,\cdot)\) \(-1\) \(1\) \(e\left(\frac{59}{70}\right)\) \(e\left(\frac{61}{210}\right)\) \(e\left(\frac{24}{35}\right)\) \(e\left(\frac{13}{210}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{37}{70}\right)\) \(e\left(\frac{61}{105}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{41}{42}\right)\)
\(\chi_{473}(148,\cdot)\) \(-1\) \(1\) \(e\left(\frac{43}{70}\right)\) \(e\left(\frac{137}{210}\right)\) \(e\left(\frac{8}{35}\right)\) \(e\left(\frac{191}{210}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{59}{70}\right)\) \(e\left(\frac{32}{105}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{37}{42}\right)\)
\(\chi_{473}(157,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{70}\right)\) \(e\left(\frac{109}{210}\right)\) \(e\left(\frac{1}{35}\right)\) \(e\left(\frac{37}{210}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{3}{70}\right)\) \(e\left(\frac{4}{105}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{23}{42}\right)\)
\(\chi_{473}(158,\cdot)\) \(-1\) \(1\) \(e\left(\frac{39}{70}\right)\) \(e\left(\frac{121}{210}\right)\) \(e\left(\frac{4}{35}\right)\) \(e\left(\frac{43}{210}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{47}{70}\right)\) \(e\left(\frac{16}{105}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{29}{42}\right)\)
\(\chi_{473}(159,\cdot)\) \(-1\) \(1\) \(e\left(\frac{33}{70}\right)\) \(e\left(\frac{97}{210}\right)\) \(e\left(\frac{33}{35}\right)\) \(e\left(\frac{31}{210}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{29}{70}\right)\) \(e\left(\frac{97}{105}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{17}{42}\right)\)
\(\chi_{473}(163,\cdot)\) \(-1\) \(1\) \(e\left(\frac{27}{70}\right)\) \(e\left(\frac{73}{210}\right)\) \(e\left(\frac{27}{35}\right)\) \(e\left(\frac{19}{210}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{11}{70}\right)\) \(e\left(\frac{73}{105}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{5}{42}\right)\)
\(\chi_{473}(190,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{70}\right)\) \(e\left(\frac{19}{210}\right)\) \(e\left(\frac{31}{35}\right)\) \(e\left(\frac{97}{210}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{23}{70}\right)\) \(e\left(\frac{19}{105}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{41}{42}\right)\)
\(\chi_{473}(191,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{70}\right)\) \(e\left(\frac{11}{210}\right)\) \(e\left(\frac{29}{35}\right)\) \(e\left(\frac{23}{210}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{17}{70}\right)\) \(e\left(\frac{11}{105}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{37}{42}\right)\)
\(\chi_{473}(192,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{70}\right)\) \(e\left(\frac{17}{210}\right)\) \(e\left(\frac{13}{35}\right)\) \(e\left(\frac{131}{210}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{39}{70}\right)\) \(e\left(\frac{17}{105}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{19}{42}\right)\)
\(\chi_{473}(201,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{70}\right)\) \(e\left(\frac{79}{210}\right)\) \(e\left(\frac{11}{35}\right)\) \(e\left(\frac{127}{210}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{33}{70}\right)\) \(e\left(\frac{79}{105}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{29}{42}\right)\)
\(\chi_{473}(202,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{70}\right)\) \(e\left(\frac{181}{210}\right)\) \(e\left(\frac{19}{35}\right)\) \(e\left(\frac{73}{210}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{57}{70}\right)\) \(e\left(\frac{76}{105}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{17}{42}\right)\)
\(\chi_{473}(218,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{70}\right)\) \(e\left(\frac{173}{210}\right)\) \(e\left(\frac{17}{35}\right)\) \(e\left(\frac{209}{210}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{51}{70}\right)\) \(e\left(\frac{68}{105}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{13}{42}\right)\)
\(\chi_{473}(234,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{70}\right)\) \(e\left(\frac{179}{210}\right)\) \(e\left(\frac{1}{35}\right)\) \(e\left(\frac{107}{210}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{3}{70}\right)\) \(e\left(\frac{74}{105}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{37}{42}\right)\)
\(\chi_{473}(235,\cdot)\) \(-1\) \(1\) \(e\left(\frac{69}{70}\right)\) \(e\left(\frac{101}{210}\right)\) \(e\left(\frac{34}{35}\right)\) \(e\left(\frac{173}{210}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{67}{70}\right)\) \(e\left(\frac{101}{105}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{19}{42}\right)\)
\(\chi_{473}(245,\cdot)\) \(-1\) \(1\) \(e\left(\frac{61}{70}\right)\) \(e\left(\frac{139}{210}\right)\) \(e\left(\frac{26}{35}\right)\) \(e\left(\frac{157}{210}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{43}{70}\right)\) \(e\left(\frac{34}{105}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{17}{42}\right)\)
\(\chi_{473}(278,\cdot)\) \(-1\) \(1\) \(e\left(\frac{41}{70}\right)\) \(e\left(\frac{59}{210}\right)\) \(e\left(\frac{6}{35}\right)\) \(e\left(\frac{47}{210}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{53}{70}\right)\) \(e\left(\frac{59}{105}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{19}{42}\right)\)
\(\chi_{473}(284,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{70}\right)\) \(e\left(\frac{43}{210}\right)\) \(e\left(\frac{2}{35}\right)\) \(e\left(\frac{109}{210}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{41}{70}\right)\) \(e\left(\frac{43}{105}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{11}{42}\right)\)
\(\chi_{473}(291,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{70}\right)\) \(e\left(\frac{197}{210}\right)\) \(e\left(\frac{23}{35}\right)\) \(e\left(\frac{11}{210}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{69}{70}\right)\) \(e\left(\frac{92}{105}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{25}{42}\right)\)