Properties

Label 8470.di
Modulus $8470$
Conductor $847$
Order $330$
Real no
Primitive no
Minimal yes
Parity odd

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([0,55,258]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(31,8470))
 
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.bd
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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 330 polynomial (not computed)

First 31 of 80 characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(9\) \(13\) \(17\) \(19\) \(23\) \(27\) \(29\) \(31\) \(37\)
\(\chi_{8470}(31,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{51}{110}\right)\) \(e\left(\frac{157}{330}\right)\) \(e\left(\frac{239}{330}\right)\) \(e\left(\frac{2}{33}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{16}{55}\right)\) \(e\left(\frac{133}{330}\right)\) \(e\left(\frac{28}{165}\right)\)
\(\chi_{8470}(201,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{13}{110}\right)\) \(e\left(\frac{191}{330}\right)\) \(e\left(\frac{7}{330}\right)\) \(e\left(\frac{10}{33}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{3}{55}\right)\) \(e\left(\frac{269}{330}\right)\) \(e\left(\frac{74}{165}\right)\)
\(\chi_{8470}(311,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{63}{110}\right)\) \(e\left(\frac{181}{330}\right)\) \(e\left(\frac{17}{330}\right)\) \(e\left(\frac{29}{33}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{23}{55}\right)\) \(e\left(\frac{229}{330}\right)\) \(e\left(\frac{109}{165}\right)\)
\(\chi_{8470}(411,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{107}{110}\right)\) \(e\left(\frac{269}{330}\right)\) \(e\left(\frac{193}{330}\right)\) \(e\left(\frac{7}{33}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{12}{55}\right)\) \(e\left(\frac{251}{330}\right)\) \(e\left(\frac{131}{165}\right)\)
\(\chi_{8470}(521,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{7}{110}\right)\) \(e\left(\frac{289}{330}\right)\) \(e\left(\frac{173}{330}\right)\) \(e\left(\frac{2}{33}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{27}{55}\right)\) \(e\left(\frac{1}{330}\right)\) \(e\left(\frac{61}{165}\right)\)
\(\chi_{8470}(621,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{19}{110}\right)\) \(e\left(\frac{203}{330}\right)\) \(e\left(\frac{61}{330}\right)\) \(e\left(\frac{7}{33}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{34}{55}\right)\) \(e\left(\frac{317}{330}\right)\) \(e\left(\frac{32}{165}\right)\)
\(\chi_{8470}(691,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{41}{110}\right)\) \(e\left(\frac{137}{330}\right)\) \(e\left(\frac{259}{330}\right)\) \(e\left(\frac{7}{33}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{1}{55}\right)\) \(e\left(\frac{53}{330}\right)\) \(e\left(\frac{98}{165}\right)\)
\(\chi_{8470}(731,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{49}{110}\right)\) \(e\left(\frac{43}{330}\right)\) \(e\left(\frac{221}{330}\right)\) \(e\left(\frac{14}{33}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{24}{55}\right)\) \(e\left(\frac{7}{330}\right)\) \(e\left(\frac{97}{165}\right)\)
\(\chi_{8470}(801,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{21}{110}\right)\) \(e\left(\frac{97}{330}\right)\) \(e\left(\frac{299}{330}\right)\) \(e\left(\frac{17}{33}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{26}{55}\right)\) \(e\left(\frac{223}{330}\right)\) \(e\left(\frac{73}{165}\right)\)
\(\chi_{8470}(1081,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{83}{110}\right)\) \(e\left(\frac{1}{330}\right)\) \(e\left(\frac{197}{330}\right)\) \(e\left(\frac{8}{33}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{53}{55}\right)\) \(e\left(\frac{169}{330}\right)\) \(e\left(\frac{79}{165}\right)\)
\(\chi_{8470}(1181,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{67}{110}\right)\) \(e\left(\frac{299}{330}\right)\) \(e\left(\frac{163}{330}\right)\) \(e\left(\frac{16}{33}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{7}{55}\right)\) \(e\left(\frac{41}{330}\right)\) \(e\left(\frac{26}{165}\right)\)
\(\chi_{8470}(1391,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{9}{110}\right)\) \(e\left(\frac{293}{330}\right)\) \(e\left(\frac{301}{330}\right)\) \(e\left(\frac{1}{33}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{19}{55}\right)\) \(e\left(\frac{17}{330}\right)\) \(e\left(\frac{47}{165}\right)\)
\(\chi_{8470}(1501,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{39}{110}\right)\) \(e\left(\frac{133}{330}\right)\) \(e\left(\frac{131}{330}\right)\) \(e\left(\frac{8}{33}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{9}{55}\right)\) \(e\left(\frac{37}{330}\right)\) \(e\left(\frac{112}{165}\right)\)
\(\chi_{8470}(1571,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{101}{110}\right)\) \(e\left(\frac{37}{330}\right)\) \(e\left(\frac{29}{330}\right)\) \(e\left(\frac{32}{33}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{36}{55}\right)\) \(e\left(\frac{313}{330}\right)\) \(e\left(\frac{118}{165}\right)\)
\(\chi_{8470}(1741,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{53}{110}\right)\) \(e\left(\frac{161}{330}\right)\) \(e\left(\frac{37}{330}\right)\) \(e\left(\frac{1}{33}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{8}{55}\right)\) \(e\left(\frac{149}{330}\right)\) \(e\left(\frac{14}{165}\right)\)
\(\chi_{8470}(1851,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{103}{110}\right)\) \(e\left(\frac{151}{330}\right)\) \(e\left(\frac{47}{330}\right)\) \(e\left(\frac{20}{33}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{28}{55}\right)\) \(e\left(\frac{109}{330}\right)\) \(e\left(\frac{49}{165}\right)\)
\(\chi_{8470}(1951,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{27}{110}\right)\) \(e\left(\frac{329}{330}\right)\) \(e\left(\frac{133}{330}\right)\) \(e\left(\frac{25}{33}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{2}{55}\right)\) \(e\left(\frac{161}{330}\right)\) \(e\left(\frac{86}{165}\right)\)
\(\chi_{8470}(2061,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{37}{110}\right)\) \(e\left(\frac{19}{330}\right)\) \(e\left(\frac{113}{330}\right)\) \(e\left(\frac{20}{33}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{17}{55}\right)\) \(e\left(\frac{241}{330}\right)\) \(e\left(\frac{16}{165}\right)\)
\(\chi_{8470}(2161,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{109}{110}\right)\) \(e\left(\frac{53}{330}\right)\) \(e\left(\frac{211}{330}\right)\) \(e\left(\frac{28}{33}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{4}{55}\right)\) \(e\left(\frac{47}{330}\right)\) \(e\left(\frac{62}{165}\right)\)
\(\chi_{8470}(2231,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{91}{110}\right)\) \(e\left(\frac{17}{330}\right)\) \(e\left(\frac{49}{330}\right)\) \(e\left(\frac{4}{33}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{21}{55}\right)\) \(e\left(\frac{233}{330}\right)\) \(e\left(\frac{23}{165}\right)\)
\(\chi_{8470}(2271,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{29}{110}\right)\) \(e\left(\frac{223}{330}\right)\) \(e\left(\frac{41}{330}\right)\) \(e\left(\frac{2}{33}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{49}{55}\right)\) \(e\left(\frac{67}{330}\right)\) \(e\left(\frac{127}{165}\right)\)
\(\chi_{8470}(2341,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{71}{110}\right)\) \(e\left(\frac{307}{330}\right)\) \(e\left(\frac{89}{330}\right)\) \(e\left(\frac{14}{33}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{46}{55}\right)\) \(e\left(\frac{73}{330}\right)\) \(e\left(\frac{163}{165}\right)\)
\(\chi_{8470}(2511,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{73}{110}\right)\) \(e\left(\frac{311}{330}\right)\) \(e\left(\frac{217}{330}\right)\) \(e\left(\frac{13}{33}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{38}{55}\right)\) \(e\left(\frac{89}{330}\right)\) \(e\left(\frac{149}{165}\right)\)
\(\chi_{8470}(2621,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{13}{110}\right)\) \(e\left(\frac{301}{330}\right)\) \(e\left(\frac{227}{330}\right)\) \(e\left(\frac{32}{33}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{3}{55}\right)\) \(e\left(\frac{49}{330}\right)\) \(e\left(\frac{19}{165}\right)\)
\(\chi_{8470}(2721,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{97}{110}\right)\) \(e\left(\frac{29}{330}\right)\) \(e\left(\frac{103}{330}\right)\) \(e\left(\frac{1}{33}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{52}{55}\right)\) \(e\left(\frac{281}{330}\right)\) \(e\left(\frac{146}{165}\right)\)
\(\chi_{8470}(2831,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{107}{110}\right)\) \(e\left(\frac{49}{330}\right)\) \(e\left(\frac{83}{330}\right)\) \(e\left(\frac{29}{33}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{12}{55}\right)\) \(e\left(\frac{31}{330}\right)\) \(e\left(\frac{76}{165}\right)\)
\(\chi_{8470}(3001,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{61}{110}\right)\) \(e\left(\frac{287}{330}\right)\) \(e\left(\frac{109}{330}\right)\) \(e\left(\frac{19}{33}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{31}{55}\right)\) \(e\left(\frac{323}{330}\right)\) \(e\left(\frac{68}{165}\right)\)
\(\chi_{8470}(3041,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{19}{110}\right)\) \(e\left(\frac{313}{330}\right)\) \(e\left(\frac{281}{330}\right)\) \(e\left(\frac{29}{33}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{34}{55}\right)\) \(e\left(\frac{97}{330}\right)\) \(e\left(\frac{142}{165}\right)\)
\(\chi_{8470}(3111,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{41}{110}\right)\) \(e\left(\frac{247}{330}\right)\) \(e\left(\frac{149}{330}\right)\) \(e\left(\frac{29}{33}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{1}{55}\right)\) \(e\left(\frac{163}{330}\right)\) \(e\left(\frac{43}{165}\right)\)
\(\chi_{8470}(3281,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{93}{110}\right)\) \(e\left(\frac{131}{330}\right)\) \(e\left(\frac{67}{330}\right)\) \(e\left(\frac{25}{33}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{13}{55}\right)\) \(e\left(\frac{29}{330}\right)\) \(e\left(\frac{119}{165}\right)\)
\(\chi_{8470}(3491,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{57}{110}\right)\) \(e\left(\frac{59}{330}\right)\) \(e\left(\frac{73}{330}\right)\) \(e\left(\frac{10}{33}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{47}{55}\right)\) \(e\left(\frac{71}{330}\right)\) \(e\left(\frac{41}{165}\right)\)