Properties

Label 8470.dr
Modulus $8470$
Conductor $4235$
Order $660$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8470, base_ring=CyclotomicField(660))
 
M = H._module
 
chi = DirichletCharacter(H, M([165,110,294]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(17,8470))
 
order = charorder(g,chi)
 
[ 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.dr
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: $\Q(\zeta_{660})$
Fixed field: Number field defined by a degree 660 polynomial (not computed)

First 29 of 160 characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(9\) \(13\) \(17\) \(19\) \(23\) \(27\) \(29\) \(31\) \(37\)
\(\chi_{8470}(17,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{53}{220}\right)\) \(e\left(\frac{161}{660}\right)\) \(e\left(\frac{101}{330}\right)\) \(e\left(\frac{35}{132}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{4}{55}\right)\) \(e\left(\frac{157}{330}\right)\) \(e\left(\frac{193}{660}\right)\)
\(\chi_{8470}(73,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{159}{220}\right)\) \(e\left(\frac{263}{660}\right)\) \(e\left(\frac{83}{330}\right)\) \(e\left(\frac{17}{132}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{12}{55}\right)\) \(e\left(\frac{31}{330}\right)\) \(e\left(\frac{139}{660}\right)\)
\(\chi_{8470}(117,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{129}{220}\right)\) \(e\left(\frac{313}{660}\right)\) \(e\left(\frac{223}{330}\right)\) \(e\left(\frac{91}{132}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{17}{55}\right)\) \(e\left(\frac{131}{330}\right)\) \(e\left(\frac{449}{660}\right)\)
\(\chi_{8470}(173,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{71}{220}\right)\) \(e\left(\frac{307}{660}\right)\) \(e\left(\frac{127}{330}\right)\) \(e\left(\frac{61}{132}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{23}{55}\right)\) \(e\left(\frac{119}{330}\right)\) \(e\left(\frac{491}{660}\right)\)
\(\chi_{8470}(227,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{109}{220}\right)\) \(e\left(\frac{53}{660}\right)\) \(e\left(\frac{23}{330}\right)\) \(e\left(\frac{23}{132}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{2}{55}\right)\) \(e\left(\frac{271}{330}\right)\) \(e\left(\frac{289}{660}\right)\)
\(\chi_{8470}(283,\cdot)\) \(-1\) \(1\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{191}{220}\right)\) \(e\left(\frac{107}{660}\right)\) \(e\left(\frac{227}{330}\right)\) \(e\left(\frac{29}{132}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{3}{55}\right)\) \(e\left(\frac{49}{330}\right)\) \(e\left(\frac{571}{660}\right)\)
\(\chi_{8470}(327,\cdot)\) \(-1\) \(1\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{41}{220}\right)\) \(e\left(\frac{577}{660}\right)\) \(e\left(\frac{157}{330}\right)\) \(e\left(\frac{91}{132}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{28}{55}\right)\) \(e\left(\frac{329}{330}\right)\) \(e\left(\frac{581}{660}\right)\)
\(\chi_{8470}(437,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{161}{220}\right)\) \(e\left(\frac{377}{660}\right)\) \(e\left(\frac{257}{330}\right)\) \(e\left(\frac{59}{132}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{8}{55}\right)\) \(e\left(\frac{259}{330}\right)\) \(e\left(\frac{1}{660}\right)\)
\(\chi_{8470}(453,\cdot)\) \(-1\) \(1\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{47}{220}\right)\) \(e\left(\frac{259}{660}\right)\) \(e\left(\frac{19}{330}\right)\) \(e\left(\frac{85}{132}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{16}{55}\right)\) \(e\left(\frac{23}{330}\right)\) \(e\left(\frac{167}{660}\right)\)
\(\chi_{8470}(523,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{63}{220}\right)\) \(e\left(\frac{511}{660}\right)\) \(e\left(\frac{91}{330}\right)\) \(e\left(\frac{25}{132}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{39}{55}\right)\) \(e\left(\frac{197}{330}\right)\) \(e\left(\frac{383}{660}\right)\)
\(\chi_{8470}(563,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{87}{220}\right)\) \(e\left(\frac{119}{660}\right)\) \(e\left(\frac{89}{330}\right)\) \(e\left(\frac{89}{132}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{46}{55}\right)\) \(e\left(\frac{73}{330}\right)\) \(e\left(\frac{487}{660}\right)\)
\(\chi_{8470}(607,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{37}{220}\right)\) \(e\left(\frac{349}{660}\right)\) \(e\left(\frac{139}{330}\right)\) \(e\left(\frac{7}{132}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{36}{55}\right)\) \(e\left(\frac{203}{330}\right)\) \(e\left(\frac{197}{660}\right)\)
\(\chi_{8470}(633,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{3}{220}\right)\) \(e\left(\frac{611}{660}\right)\) \(e\left(\frac{41}{330}\right)\) \(e\left(\frac{41}{132}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{49}{55}\right)\) \(e\left(\frac{67}{330}\right)\) \(e\left(\frac{343}{660}\right)\)
\(\chi_{8470}(677,\cdot)\) \(-1\) \(1\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{133}{220}\right)\) \(e\left(\frac{541}{660}\right)\) \(e\left(\frac{241}{330}\right)\) \(e\left(\frac{43}{132}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{9}{55}\right)\) \(e\left(\frac{257}{330}\right)\) \(e\left(\frac{173}{660}\right)\)
\(\chi_{8470}(733,\cdot)\) \(-1\) \(1\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{39}{220}\right)\) \(e\left(\frac{463}{660}\right)\) \(e\left(\frac{313}{330}\right)\) \(e\left(\frac{49}{132}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{32}{55}\right)\) \(e\left(\frac{101}{330}\right)\) \(e\left(\frac{59}{660}\right)\)
\(\chi_{8470}(787,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{73}{220}\right)\) \(e\left(\frac{641}{660}\right)\) \(e\left(\frac{191}{330}\right)\) \(e\left(\frac{59}{132}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{19}{55}\right)\) \(e\left(\frac{127}{330}\right)\) \(e\left(\frac{133}{660}\right)\)
\(\chi_{8470}(843,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{19}{220}\right)\) \(e\left(\frac{203}{660}\right)\) \(e\left(\frac{113}{330}\right)\) \(e\left(\frac{113}{132}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{17}{55}\right)\) \(e\left(\frac{241}{330}\right)\) \(e\left(\frac{559}{660}\right)\)
\(\chi_{8470}(943,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{31}{220}\right)\) \(e\left(\frac{7}{660}\right)\) \(e\left(\frac{277}{330}\right)\) \(e\left(\frac{13}{132}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{48}{55}\right)\) \(e\left(\frac{179}{330}\right)\) \(e\left(\frac{611}{660}\right)\)
\(\chi_{8470}(997,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{189}{220}\right)\) \(e\left(\frac{653}{660}\right)\) \(e\left(\frac{53}{330}\right)\) \(e\left(\frac{119}{132}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{7}{55}\right)\) \(e\left(\frac{151}{330}\right)\) \(e\left(\frac{49}{660}\right)\)
\(\chi_{8470}(1053,\cdot)\) \(-1\) \(1\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{151}{220}\right)\) \(e\left(\frac{467}{660}\right)\) \(e\left(\frac{47}{330}\right)\) \(e\left(\frac{113}{132}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{28}{55}\right)\) \(e\left(\frac{109}{330}\right)\) \(e\left(\frac{31}{660}\right)\)
\(\chi_{8470}(1097,\cdot)\) \(-1\) \(1\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{1}{220}\right)\) \(e\left(\frac{277}{660}\right)\) \(e\left(\frac{307}{330}\right)\) \(e\left(\frac{43}{132}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{53}{55}\right)\) \(e\left(\frac{59}{330}\right)\) \(e\left(\frac{41}{660}\right)\)
\(\chi_{8470}(1223,\cdot)\) \(-1\) \(1\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{107}{220}\right)\) \(e\left(\frac{379}{660}\right)\) \(e\left(\frac{289}{330}\right)\) \(e\left(\frac{25}{132}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{6}{55}\right)\) \(e\left(\frac{263}{330}\right)\) \(e\left(\frac{647}{660}\right)\)
\(\chi_{8470}(1293,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{83}{220}\right)\) \(e\left(\frac{331}{660}\right)\) \(e\left(\frac{181}{330}\right)\) \(e\left(\frac{49}{132}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{54}{55}\right)\) \(e\left(\frac{167}{330}\right)\) \(e\left(\frac{323}{660}\right)\)
\(\chi_{8470}(1333,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{147}{220}\right)\) \(e\left(\frac{239}{660}\right)\) \(e\left(\frac{29}{330}\right)\) \(e\left(\frac{29}{132}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{36}{55}\right)\) \(e\left(\frac{313}{330}\right)\) \(e\left(\frac{307}{660}\right)\)
\(\chi_{8470}(1377,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{97}{220}\right)\) \(e\left(\frac{469}{660}\right)\) \(e\left(\frac{79}{330}\right)\) \(e\left(\frac{79}{132}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{26}{55}\right)\) \(e\left(\frac{113}{330}\right)\) \(e\left(\frac{17}{660}\right)\)
\(\chi_{8470}(1403,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{23}{220}\right)\) \(e\left(\frac{431}{660}\right)\) \(e\left(\frac{131}{330}\right)\) \(e\left(\frac{65}{132}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{9}{55}\right)\) \(e\left(\frac{37}{330}\right)\) \(e\left(\frac{283}{660}\right)\)
\(\chi_{8470}(1447,\cdot)\) \(-1\) \(1\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{153}{220}\right)\) \(e\left(\frac{361}{660}\right)\) \(e\left(\frac{1}{330}\right)\) \(e\left(\frac{67}{132}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{24}{55}\right)\) \(e\left(\frac{227}{330}\right)\) \(e\left(\frac{113}{660}\right)\)
\(\chi_{8470}(1487,\cdot)\) \(-1\) \(1\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{137}{220}\right)\) \(e\left(\frac{329}{660}\right)\) \(e\left(\frac{149}{330}\right)\) \(e\left(\frac{83}{132}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{1}{55}\right)\) \(e\left(\frac{163}{330}\right)\) \(e\left(\frac{337}{660}\right)\)
\(\chi_{8470}(1503,\cdot)\) \(-1\) \(1\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{119}{220}\right)\) \(e\left(\frac{403}{660}\right)\) \(e\left(\frac{13}{330}\right)\) \(e\left(\frac{13}{132}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{37}{55}\right)\) \(e\left(\frac{311}{330}\right)\) \(e\left(\frac{479}{660}\right)\)