Properties

Label 8470.dj
Modulus $8470$
Conductor $847$
Order $330$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(8470, base_ring=CyclotomicField(330))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,275,327]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(61,8470))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(8470\)
Conductor: \(847\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from 847.be
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 330 polynomial (not computed)

First 31 of 80 characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(9\) \(13\) \(17\) \(19\) \(23\) \(27\) \(29\) \(31\) \(37\)
\(\chi_{8470}(61,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{32}{55}\right)\) \(e\left(\frac{64}{165}\right)\) \(e\left(\frac{68}{165}\right)\) \(e\left(\frac{1}{33}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{93}{110}\right)\) \(e\left(\frac{17}{330}\right)\) \(e\left(\frac{47}{165}\right)\)
\(\chi_{8470}(101,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{43}{55}\right)\) \(e\left(\frac{86}{165}\right)\) \(e\left(\frac{112}{165}\right)\) \(e\left(\frac{23}{33}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{27}{110}\right)\) \(e\left(\frac{193}{330}\right)\) \(e\left(\frac{58}{165}\right)\)
\(\chi_{8470}(171,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{17}{55}\right)\) \(e\left(\frac{89}{165}\right)\) \(e\left(\frac{43}{165}\right)\) \(e\left(\frac{5}{33}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{3}{110}\right)\) \(e\left(\frac{217}{330}\right)\) \(e\left(\frac{37}{165}\right)\)
\(\chi_{8470}(271,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{6}{55}\right)\) \(e\left(\frac{67}{165}\right)\) \(e\left(\frac{164}{165}\right)\) \(e\left(\frac{16}{33}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{69}{110}\right)\) \(e\left(\frac{41}{330}\right)\) \(e\left(\frac{26}{165}\right)\)
\(\chi_{8470}(381,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{1}{55}\right)\) \(e\left(\frac{2}{165}\right)\) \(e\left(\frac{64}{165}\right)\) \(e\left(\frac{32}{33}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{39}{110}\right)\) \(e\left(\frac{181}{330}\right)\) \(e\left(\frac{151}{165}\right)\)
\(\chi_{8470}(591,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{19}{55}\right)\) \(e\left(\frac{38}{165}\right)\) \(e\left(\frac{61}{165}\right)\) \(e\left(\frac{14}{33}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{81}{110}\right)\) \(e\left(\frac{139}{330}\right)\) \(e\left(\frac{64}{165}\right)\)
\(\chi_{8470}(761,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{48}{55}\right)\) \(e\left(\frac{151}{165}\right)\) \(e\left(\frac{47}{165}\right)\) \(e\left(\frac{7}{33}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{57}{110}\right)\) \(e\left(\frac{53}{330}\right)\) \(e\left(\frac{98}{165}\right)\)
\(\chi_{8470}(831,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{37}{55}\right)\) \(e\left(\frac{19}{165}\right)\) \(e\left(\frac{113}{165}\right)\) \(e\left(\frac{7}{33}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{13}{110}\right)\) \(e\left(\frac{317}{330}\right)\) \(e\left(\frac{32}{165}\right)\)
\(\chi_{8470}(871,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{3}{55}\right)\) \(e\left(\frac{116}{165}\right)\) \(e\left(\frac{82}{165}\right)\) \(e\left(\frac{8}{33}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{7}{110}\right)\) \(e\left(\frac{103}{330}\right)\) \(e\left(\frac{13}{165}\right)\)
\(\chi_{8470}(1041,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{26}{55}\right)\) \(e\left(\frac{52}{165}\right)\) \(e\left(\frac{14}{165}\right)\) \(e\left(\frac{7}{33}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{79}{110}\right)\) \(e\left(\frac{251}{330}\right)\) \(e\left(\frac{131}{165}\right)\)
\(\chi_{8470}(1151,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{21}{55}\right)\) \(e\left(\frac{152}{165}\right)\) \(e\left(\frac{79}{165}\right)\) \(e\left(\frac{23}{33}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{49}{110}\right)\) \(e\left(\frac{61}{330}\right)\) \(e\left(\frac{91}{165}\right)\)
\(\chi_{8470}(1251,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{34}{55}\right)\) \(e\left(\frac{13}{165}\right)\) \(e\left(\frac{86}{165}\right)\) \(e\left(\frac{10}{33}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{61}{110}\right)\) \(e\left(\frac{269}{330}\right)\) \(e\left(\frac{74}{165}\right)\)
\(\chi_{8470}(1361,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{9}{55}\right)\) \(e\left(\frac{128}{165}\right)\) \(e\left(\frac{136}{165}\right)\) \(e\left(\frac{2}{33}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{21}{110}\right)\) \(e\left(\frac{199}{330}\right)\) \(e\left(\frac{94}{165}\right)\)
\(\chi_{8470}(1531,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{8}{55}\right)\) \(e\left(\frac{16}{165}\right)\) \(e\left(\frac{17}{165}\right)\) \(e\left(\frac{25}{33}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{37}{110}\right)\) \(e\left(\frac{293}{330}\right)\) \(e\left(\frac{53}{165}\right)\)
\(\chi_{8470}(1601,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{42}{55}\right)\) \(e\left(\frac{139}{165}\right)\) \(e\left(\frac{158}{165}\right)\) \(e\left(\frac{13}{33}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{43}{110}\right)\) \(e\left(\frac{287}{330}\right)\) \(e\left(\frac{17}{165}\right)\)
\(\chi_{8470}(1641,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{18}{55}\right)\) \(e\left(\frac{146}{165}\right)\) \(e\left(\frac{52}{165}\right)\) \(e\left(\frac{26}{33}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{97}{110}\right)\) \(e\left(\frac{13}{330}\right)\) \(e\left(\frac{133}{165}\right)\)
\(\chi_{8470}(1711,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{27}{55}\right)\) \(e\left(\frac{164}{165}\right)\) \(e\left(\frac{133}{165}\right)\) \(e\left(\frac{17}{33}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{63}{110}\right)\) \(e\left(\frac{157}{330}\right)\) \(e\left(\frac{7}{165}\right)\)
\(\chi_{8470}(1811,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{46}{55}\right)\) \(e\left(\frac{37}{165}\right)\) \(e\left(\frac{29}{165}\right)\) \(e\left(\frac{31}{33}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{89}{110}\right)\) \(e\left(\frac{131}{330}\right)\) \(e\left(\frac{71}{165}\right)\)
\(\chi_{8470}(1921,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{41}{55}\right)\) \(e\left(\frac{137}{165}\right)\) \(e\left(\frac{94}{165}\right)\) \(e\left(\frac{14}{33}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{59}{110}\right)\) \(e\left(\frac{271}{330}\right)\) \(e\left(\frac{31}{165}\right)\)
\(\chi_{8470}(2021,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{24}{55}\right)\) \(e\left(\frac{103}{165}\right)\) \(e\left(\frac{161}{165}\right)\) \(e\left(\frac{31}{33}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{1}{110}\right)\) \(e\left(\frac{329}{330}\right)\) \(e\left(\frac{104}{165}\right)\)
\(\chi_{8470}(2131,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{54}{55}\right)\) \(e\left(\frac{53}{165}\right)\) \(e\left(\frac{46}{165}\right)\) \(e\left(\frac{23}{33}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{71}{110}\right)\) \(e\left(\frac{259}{330}\right)\) \(e\left(\frac{124}{165}\right)\)
\(\chi_{8470}(2301,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{23}{55}\right)\) \(e\left(\frac{46}{165}\right)\) \(e\left(\frac{152}{165}\right)\) \(e\left(\frac{10}{33}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{17}{110}\right)\) \(e\left(\frac{203}{330}\right)\) \(e\left(\frac{8}{165}\right)\)
\(\chi_{8470}(2371,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{47}{55}\right)\) \(e\left(\frac{94}{165}\right)\) \(e\left(\frac{38}{165}\right)\) \(e\left(\frac{19}{33}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{73}{110}\right)\) \(e\left(\frac{257}{330}\right)\) \(e\left(\frac{2}{165}\right)\)
\(\chi_{8470}(2481,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{32}{55}\right)\) \(e\left(\frac{119}{165}\right)\) \(e\left(\frac{13}{165}\right)\) \(e\left(\frac{23}{33}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{93}{110}\right)\) \(e\left(\frac{127}{330}\right)\) \(e\left(\frac{157}{165}\right)\)
\(\chi_{8470}(2691,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{6}{55}\right)\) \(e\left(\frac{122}{165}\right)\) \(e\left(\frac{109}{165}\right)\) \(e\left(\frac{5}{33}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{69}{110}\right)\) \(e\left(\frac{151}{330}\right)\) \(e\left(\frac{136}{165}\right)\)
\(\chi_{8470}(2791,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{14}{55}\right)\) \(e\left(\frac{28}{165}\right)\) \(e\left(\frac{71}{165}\right)\) \(e\left(\frac{19}{33}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{51}{110}\right)\) \(e\left(\frac{59}{330}\right)\) \(e\left(\frac{134}{165}\right)\)
\(\chi_{8470}(3071,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{38}{55}\right)\) \(e\left(\frac{76}{165}\right)\) \(e\left(\frac{122}{165}\right)\) \(e\left(\frac{28}{33}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{107}{110}\right)\) \(e\left(\frac{113}{330}\right)\) \(e\left(\frac{128}{165}\right)\)
\(\chi_{8470}(3141,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{52}{55}\right)\) \(e\left(\frac{49}{165}\right)\) \(e\left(\frac{83}{165}\right)\) \(e\left(\frac{25}{33}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{103}{110}\right)\) \(e\left(\frac{227}{330}\right)\) \(e\left(\frac{152}{165}\right)\)
\(\chi_{8470}(3181,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{48}{55}\right)\) \(e\left(\frac{41}{165}\right)\) \(e\left(\frac{157}{165}\right)\) \(e\left(\frac{29}{33}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{57}{110}\right)\) \(e\left(\frac{163}{330}\right)\) \(e\left(\frac{43}{165}\right)\)
\(\chi_{8470}(3251,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{37}{55}\right)\) \(e\left(\frac{74}{165}\right)\) \(e\left(\frac{58}{165}\right)\) \(e\left(\frac{29}{33}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{13}{110}\right)\) \(e\left(\frac{97}{330}\right)\) \(e\left(\frac{142}{165}\right)\)
\(\chi_{8470}(3351,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{31}{55}\right)\) \(e\left(\frac{7}{165}\right)\) \(e\left(\frac{59}{165}\right)\) \(e\left(\frac{13}{33}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{109}{110}\right)\) \(e\left(\frac{221}{330}\right)\) \(e\left(\frac{116}{165}\right)\)