Properties

Label 8470.dp
Modulus $8470$
Conductor $4235$
Order $660$
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(660))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([165,220,378]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(107,8470))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(8470\)
Conductor: \(4235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(660\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from 4235.do
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: $\Q(\zeta_{660})$
Fixed field: Number field defined by a degree 660 polynomial (not computed)

First 31 of 160 characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(9\) \(13\) \(17\) \(19\) \(23\) \(27\) \(29\) \(31\) \(37\)
\(\chi_{8470}(107,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{131}{220}\right)\) \(e\left(\frac{427}{660}\right)\) \(e\left(\frac{116}{165}\right)\) \(e\left(\frac{67}{132}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{13}{55}\right)\) \(e\left(\frac{97}{165}\right)\) \(e\left(\frac{641}{660}\right)\)
\(\chi_{8470}(123,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{37}{220}\right)\) \(e\left(\frac{569}{660}\right)\) \(e\left(\frac{97}{165}\right)\) \(e\left(\frac{29}{132}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{36}{55}\right)\) \(e\left(\frac{74}{165}\right)\) \(e\left(\frac{307}{660}\right)\)
\(\chi_{8470}(193,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{133}{220}\right)\) \(e\left(\frac{101}{660}\right)\) \(e\left(\frac{148}{165}\right)\) \(e\left(\frac{65}{132}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{9}{55}\right)\) \(e\left(\frac{101}{165}\right)\) \(e\left(\frac{283}{660}\right)\)
\(\chi_{8470}(277,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{27}{220}\right)\) \(e\left(\frac{659}{660}\right)\) \(e\left(\frac{157}{165}\right)\) \(e\left(\frac{83}{132}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{1}{55}\right)\) \(e\left(\frac{164}{165}\right)\) \(e\left(\frac{337}{660}\right)\)
\(\chi_{8470}(303,\cdot)\) \(1\) \(1\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{73}{220}\right)\) \(e\left(\frac{421}{660}\right)\) \(e\left(\frac{68}{165}\right)\) \(e\left(\frac{37}{132}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{19}{55}\right)\) \(e\left(\frac{91}{165}\right)\) \(e\left(\frac{23}{660}\right)\)
\(\chi_{8470}(347,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{203}{220}\right)\) \(e\left(\frac{131}{660}\right)\) \(e\left(\frac{58}{165}\right)\) \(e\left(\frac{83}{132}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{34}{55}\right)\) \(e\left(\frac{131}{165}\right)\) \(e\left(\frac{73}{660}\right)\)
\(\chi_{8470}(387,\cdot)\) \(1\) \(1\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{67}{220}\right)\) \(e\left(\frac{79}{660}\right)\) \(e\left(\frac{137}{165}\right)\) \(e\left(\frac{43}{132}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{31}{55}\right)\) \(e\left(\frac{79}{165}\right)\) \(e\left(\frac{437}{660}\right)\)
\(\chi_{8470}(513,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{189}{220}\right)\) \(e\left(\frac{433}{660}\right)\) \(e\left(\frac{164}{165}\right)\) \(e\left(\frac{97}{132}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{7}{55}\right)\) \(e\left(\frac{103}{165}\right)\) \(e\left(\frac{599}{660}\right)\)
\(\chi_{8470}(557,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{159}{220}\right)\) \(e\left(\frac{263}{660}\right)\) \(e\left(\frac{124}{165}\right)\) \(e\left(\frac{83}{132}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{12}{55}\right)\) \(e\left(\frac{98}{165}\right)\) \(e\left(\frac{469}{660}\right)\)
\(\chi_{8470}(613,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{1}{220}\right)\) \(e\left(\frac{497}{660}\right)\) \(e\left(\frac{16}{165}\right)\) \(e\left(\frac{65}{132}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{53}{55}\right)\) \(e\left(\frac{2}{165}\right)\) \(e\left(\frac{151}{660}\right)\)
\(\chi_{8470}(667,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{139}{220}\right)\) \(e\left(\frac{223}{660}\right)\) \(e\left(\frac{134}{165}\right)\) \(e\left(\frac{103}{132}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{52}{55}\right)\) \(e\left(\frac{58}{165}\right)\) \(e\left(\frac{89}{660}\right)\)
\(\chi_{8470}(767,\cdot)\) \(1\) \(1\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{191}{220}\right)\) \(e\left(\frac{107}{660}\right)\) \(e\left(\frac{31}{165}\right)\) \(e\left(\frac{95}{132}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{3}{55}\right)\) \(e\left(\frac{107}{165}\right)\) \(e\left(\frac{241}{660}\right)\)
\(\chi_{8470}(877,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{91}{220}\right)\) \(e\left(\frac{127}{660}\right)\) \(e\left(\frac{26}{165}\right)\) \(e\left(\frac{19}{132}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{38}{55}\right)\) \(e\left(\frac{127}{165}\right)\) \(e\left(\frac{101}{660}\right)\)
\(\chi_{8470}(893,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{97}{220}\right)\) \(e\left(\frac{29}{660}\right)\) \(e\left(\frac{67}{165}\right)\) \(e\left(\frac{101}{132}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{26}{55}\right)\) \(e\left(\frac{29}{165}\right)\) \(e\left(\frac{127}{660}\right)\)
\(\chi_{8470}(963,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{153}{220}\right)\) \(e\left(\frac{581}{660}\right)\) \(e\left(\frac{28}{165}\right)\) \(e\left(\frac{89}{132}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{24}{55}\right)\) \(e\left(\frac{86}{165}\right)\) \(e\left(\frac{223}{660}\right)\)
\(\chi_{8470}(1003,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{137}{220}\right)\) \(e\left(\frac{109}{660}\right)\) \(e\left(\frac{47}{165}\right)\) \(e\left(\frac{61}{132}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{1}{55}\right)\) \(e\left(\frac{109}{165}\right)\) \(e\left(\frac{227}{660}\right)\)
\(\chi_{8470}(1047,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{87}{220}\right)\) \(e\left(\frac{119}{660}\right)\) \(e\left(\frac{127}{165}\right)\) \(e\left(\frac{23}{132}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{46}{55}\right)\) \(e\left(\frac{119}{165}\right)\) \(e\left(\frac{157}{660}\right)\)
\(\chi_{8470}(1073,\cdot)\) \(1\) \(1\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{93}{220}\right)\) \(e\left(\frac{241}{660}\right)\) \(e\left(\frac{113}{165}\right)\) \(e\left(\frac{61}{132}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{34}{55}\right)\) \(e\left(\frac{76}{165}\right)\) \(e\left(\frac{623}{660}\right)\)
\(\chi_{8470}(1117,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{3}{220}\right)\) \(e\left(\frac{611}{660}\right)\) \(e\left(\frac{103}{165}\right)\) \(e\left(\frac{107}{132}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{49}{55}\right)\) \(e\left(\frac{116}{165}\right)\) \(e\left(\frac{13}{660}\right)\)
\(\chi_{8470}(1157,\cdot)\) \(1\) \(1\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{127}{220}\right)\) \(e\left(\frac{199}{660}\right)\) \(e\left(\frac{107}{165}\right)\) \(e\left(\frac{115}{132}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{21}{55}\right)\) \(e\left(\frac{34}{165}\right)\) \(e\left(\frac{257}{660}\right)\)
\(\chi_{8470}(1173,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{69}{220}\right)\) \(e\left(\frac{413}{660}\right)\) \(e\left(\frac{4}{165}\right)\) \(e\left(\frac{41}{132}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{27}{55}\right)\) \(e\left(\frac{83}{165}\right)\) \(e\left(\frac{79}{660}\right)\)
\(\chi_{8470}(1227,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{163}{220}\right)\) \(e\left(\frac{271}{660}\right)\) \(e\left(\frac{23}{165}\right)\) \(e\left(\frac{79}{132}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{4}{55}\right)\) \(e\left(\frac{106}{165}\right)\) \(e\left(\frac{413}{660}\right)\)
\(\chi_{8470}(1283,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{49}{220}\right)\) \(e\left(\frac{373}{660}\right)\) \(e\left(\frac{14}{165}\right)\) \(e\left(\frac{61}{132}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{12}{55}\right)\) \(e\left(\frac{43}{165}\right)\) \(e\left(\frac{359}{660}\right)\)
\(\chi_{8470}(1327,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{19}{220}\right)\) \(e\left(\frac{203}{660}\right)\) \(e\left(\frac{139}{165}\right)\) \(e\left(\frac{47}{132}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{17}{55}\right)\) \(e\left(\frac{38}{165}\right)\) \(e\left(\frac{229}{660}\right)\)
\(\chi_{8470}(1383,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{181}{220}\right)\) \(e\left(\frac{197}{660}\right)\) \(e\left(\frac{91}{165}\right)\) \(e\left(\frac{17}{132}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{23}{55}\right)\) \(e\left(\frac{32}{165}\right)\) \(e\left(\frac{271}{660}\right)\)
\(\chi_{8470}(1437,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{219}{220}\right)\) \(e\left(\frac{163}{660}\right)\) \(e\left(\frac{149}{165}\right)\) \(e\left(\frac{67}{132}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{2}{55}\right)\) \(e\left(\frac{163}{165}\right)\) \(e\left(\frac{509}{660}\right)\)
\(\chi_{8470}(1493,\cdot)\) \(1\) \(1\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{81}{220}\right)\) \(e\left(\frac{217}{660}\right)\) \(e\left(\frac{86}{165}\right)\) \(e\left(\frac{73}{132}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{3}{55}\right)\) \(e\left(\frac{52}{165}\right)\) \(e\left(\frac{131}{660}\right)\)
\(\chi_{8470}(1537,\cdot)\) \(1\) \(1\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{151}{220}\right)\) \(e\left(\frac{467}{660}\right)\) \(e\left(\frac{106}{165}\right)\) \(e\left(\frac{47}{132}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{28}{55}\right)\) \(e\left(\frac{137}{165}\right)\) \(e\left(\frac{361}{660}\right)\)
\(\chi_{8470}(1647,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{51}{220}\right)\) \(e\left(\frac{487}{660}\right)\) \(e\left(\frac{101}{165}\right)\) \(e\left(\frac{103}{132}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{8}{55}\right)\) \(e\left(\frac{157}{165}\right)\) \(e\left(\frac{221}{660}\right)\)
\(\chi_{8470}(1663,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{157}{220}\right)\) \(e\left(\frac{149}{660}\right)\) \(e\left(\frac{37}{165}\right)\) \(e\left(\frac{41}{132}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{16}{55}\right)\) \(e\left(\frac{149}{165}\right)\) \(e\left(\frac{607}{660}\right)\)
\(\chi_{8470}(1733,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{173}{220}\right)\) \(e\left(\frac{401}{660}\right)\) \(e\left(\frac{73}{165}\right)\) \(e\left(\frac{113}{132}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{39}{55}\right)\) \(e\left(\frac{71}{165}\right)\) \(e\left(\frac{163}{660}\right)\)