Properties

Label 8470.dl
Modulus $8470$
Conductor $4235$
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([165,275,249]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(19,8470))
 
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: \(330\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from 4235.dj
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 \(-1\) \(1\) \(3\) \(9\) \(13\) \(17\) \(19\) \(23\) \(27\) \(29\) \(31\) \(37\)
\(\chi_{8470}(19,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{23}{110}\right)\) \(e\left(\frac{101}{330}\right)\) \(e\left(\frac{131}{165}\right)\) \(e\left(\frac{65}{66}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{91}{110}\right)\) \(e\left(\frac{239}{330}\right)\) \(e\left(\frac{283}{330}\right)\)
\(\chi_{8470}(129,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{83}{110}\right)\) \(e\left(\frac{1}{330}\right)\) \(e\left(\frac{16}{165}\right)\) \(e\left(\frac{49}{66}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{51}{110}\right)\) \(e\left(\frac{169}{330}\right)\) \(e\left(\frac{323}{330}\right)\)
\(\chi_{8470}(299,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{1}{110}\right)\) \(e\left(\frac{167}{330}\right)\) \(e\left(\frac{32}{165}\right)\) \(e\left(\frac{65}{66}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{47}{110}\right)\) \(e\left(\frac{173}{330}\right)\) \(e\left(\frac{151}{330}\right)\)
\(\chi_{8470}(369,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{79}{110}\right)\) \(e\left(\frac{323}{330}\right)\) \(e\left(\frac{53}{165}\right)\) \(e\left(\frac{53}{66}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{83}{110}\right)\) \(e\left(\frac{137}{330}\right)\) \(e\left(\frac{49}{330}\right)\)
\(\chi_{8470}(409,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{21}{110}\right)\) \(e\left(\frac{97}{330}\right)\) \(e\left(\frac{67}{165}\right)\) \(e\left(\frac{1}{66}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{107}{110}\right)\) \(e\left(\frac{223}{330}\right)\) \(e\left(\frac{311}{330}\right)\)
\(\chi_{8470}(479,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{49}{110}\right)\) \(e\left(\frac{43}{330}\right)\) \(e\left(\frac{28}{165}\right)\) \(e\left(\frac{61}{66}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{103}{110}\right)\) \(e\left(\frac{7}{330}\right)\) \(e\left(\frac{29}{330}\right)\)
\(\chi_{8470}(579,\cdot)\) \(1\) \(1\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{17}{110}\right)\) \(e\left(\frac{89}{330}\right)\) \(e\left(\frac{104}{165}\right)\) \(e\left(\frac{5}{66}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{29}{110}\right)\) \(e\left(\frac{191}{330}\right)\) \(e\left(\frac{37}{330}\right)\)
\(\chi_{8470}(689,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{7}{110}\right)\) \(e\left(\frac{289}{330}\right)\) \(e\left(\frac{4}{165}\right)\) \(e\left(\frac{37}{66}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{109}{110}\right)\) \(e\left(\frac{1}{330}\right)\) \(e\left(\frac{287}{330}\right)\)
\(\chi_{8470}(789,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{3}{110}\right)\) \(e\left(\frac{281}{330}\right)\) \(e\left(\frac{41}{165}\right)\) \(e\left(\frac{41}{66}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{31}{110}\right)\) \(e\left(\frac{299}{330}\right)\) \(e\left(\frac{13}{330}\right)\)
\(\chi_{8470}(899,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{63}{110}\right)\) \(e\left(\frac{181}{330}\right)\) \(e\left(\frac{91}{165}\right)\) \(e\left(\frac{25}{66}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{101}{110}\right)\) \(e\left(\frac{229}{330}\right)\) \(e\left(\frac{53}{330}\right)\)
\(\chi_{8470}(1069,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{31}{110}\right)\) \(e\left(\frac{227}{330}\right)\) \(e\left(\frac{2}{165}\right)\) \(e\left(\frac{35}{66}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{27}{110}\right)\) \(e\left(\frac{83}{330}\right)\) \(e\left(\frac{61}{330}\right)\)
\(\chi_{8470}(1139,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{89}{110}\right)\) \(e\left(\frac{233}{330}\right)\) \(e\left(\frac{98}{165}\right)\) \(e\left(\frac{65}{66}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{3}{110}\right)\) \(e\left(\frac{107}{330}\right)\) \(e\left(\frac{19}{330}\right)\)
\(\chi_{8470}(1179,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{51}{110}\right)\) \(e\left(\frac{157}{330}\right)\) \(e\left(\frac{37}{165}\right)\) \(e\left(\frac{37}{66}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{87}{110}\right)\) \(e\left(\frac{133}{330}\right)\) \(e\left(\frac{221}{330}\right)\)
\(\chi_{8470}(1249,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{59}{110}\right)\) \(e\left(\frac{283}{330}\right)\) \(e\left(\frac{73}{165}\right)\) \(e\left(\frac{7}{66}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{23}{110}\right)\) \(e\left(\frac{307}{330}\right)\) \(e\left(\frac{329}{330}\right)\)
\(\chi_{8470}(1349,\cdot)\) \(1\) \(1\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{57}{110}\right)\) \(e\left(\frac{59}{330}\right)\) \(e\left(\frac{119}{165}\right)\) \(e\left(\frac{53}{66}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{39}{110}\right)\) \(e\left(\frac{71}{330}\right)\) \(e\left(\frac{247}{330}\right)\)
\(\chi_{8470}(1459,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{47}{110}\right)\) \(e\left(\frac{259}{330}\right)\) \(e\left(\frac{19}{165}\right)\) \(e\left(\frac{19}{66}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{9}{110}\right)\) \(e\left(\frac{211}{330}\right)\) \(e\left(\frac{167}{330}\right)\)
\(\chi_{8470}(1559,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{93}{110}\right)\) \(e\left(\frac{131}{330}\right)\) \(e\left(\frac{116}{165}\right)\) \(e\left(\frac{17}{66}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{81}{110}\right)\) \(e\left(\frac{29}{330}\right)\) \(e\left(\frac{73}{330}\right)\)
\(\chi_{8470}(1669,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{43}{110}\right)\) \(e\left(\frac{31}{330}\right)\) \(e\left(\frac{1}{165}\right)\) \(e\left(\frac{1}{66}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{41}{110}\right)\) \(e\left(\frac{289}{330}\right)\) \(e\left(\frac{113}{330}\right)\)
\(\chi_{8470}(1839,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{61}{110}\right)\) \(e\left(\frac{287}{330}\right)\) \(e\left(\frac{137}{165}\right)\) \(e\left(\frac{5}{66}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{7}{110}\right)\) \(e\left(\frac{323}{330}\right)\) \(e\left(\frac{301}{330}\right)\)
\(\chi_{8470}(1949,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{81}{110}\right)\) \(e\left(\frac{217}{330}\right)\) \(e\left(\frac{7}{165}\right)\) \(e\left(\frac{7}{66}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{67}{110}\right)\) \(e\left(\frac{43}{330}\right)\) \(e\left(\frac{131}{330}\right)\)
\(\chi_{8470}(2019,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{69}{110}\right)\) \(e\left(\frac{193}{330}\right)\) \(e\left(\frac{118}{165}\right)\) \(e\left(\frac{19}{66}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{53}{110}\right)\) \(e\left(\frac{277}{330}\right)\) \(e\left(\frac{299}{330}\right)\)
\(\chi_{8470}(2119,\cdot)\) \(1\) \(1\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{97}{110}\right)\) \(e\left(\frac{29}{330}\right)\) \(e\left(\frac{134}{165}\right)\) \(e\left(\frac{35}{66}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{49}{110}\right)\) \(e\left(\frac{281}{330}\right)\) \(e\left(\frac{127}{330}\right)\)
\(\chi_{8470}(2229,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{87}{110}\right)\) \(e\left(\frac{229}{330}\right)\) \(e\left(\frac{34}{165}\right)\) \(e\left(\frac{1}{66}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{19}{110}\right)\) \(e\left(\frac{91}{330}\right)\) \(e\left(\frac{47}{330}\right)\)
\(\chi_{8470}(2329,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{73}{110}\right)\) \(e\left(\frac{311}{330}\right)\) \(e\left(\frac{26}{165}\right)\) \(e\left(\frac{59}{66}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{21}{110}\right)\) \(e\left(\frac{89}{330}\right)\) \(e\left(\frac{133}{330}\right)\)
\(\chi_{8470}(2439,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{23}{110}\right)\) \(e\left(\frac{211}{330}\right)\) \(e\left(\frac{76}{165}\right)\) \(e\left(\frac{43}{66}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{91}{110}\right)\) \(e\left(\frac{19}{330}\right)\) \(e\left(\frac{173}{330}\right)\)
\(\chi_{8470}(2609,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{91}{110}\right)\) \(e\left(\frac{17}{330}\right)\) \(e\left(\frac{107}{165}\right)\) \(e\left(\frac{41}{66}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{97}{110}\right)\) \(e\left(\frac{233}{330}\right)\) \(e\left(\frac{211}{330}\right)\)
\(\chi_{8470}(2679,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{109}{110}\right)\) \(e\left(\frac{53}{330}\right)\) \(e\left(\frac{23}{165}\right)\) \(e\left(\frac{23}{66}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{63}{110}\right)\) \(e\left(\frac{47}{330}\right)\) \(e\left(\frac{289}{330}\right)\)
\(\chi_{8470}(2719,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{1}{110}\right)\) \(e\left(\frac{277}{330}\right)\) \(e\left(\frac{142}{165}\right)\) \(e\left(\frac{43}{66}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{47}{110}\right)\) \(e\left(\frac{283}{330}\right)\) \(e\left(\frac{41}{330}\right)\)
\(\chi_{8470}(2789,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{79}{110}\right)\) \(e\left(\frac{103}{330}\right)\) \(e\left(\frac{163}{165}\right)\) \(e\left(\frac{31}{66}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{83}{110}\right)\) \(e\left(\frac{247}{330}\right)\) \(e\left(\frac{269}{330}\right)\)
\(\chi_{8470}(2889,\cdot)\) \(1\) \(1\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{27}{110}\right)\) \(e\left(\frac{329}{330}\right)\) \(e\left(\frac{149}{165}\right)\) \(e\left(\frac{17}{66}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{59}{110}\right)\) \(e\left(\frac{161}{330}\right)\) \(e\left(\frac{7}{330}\right)\)
\(\chi_{8470}(2999,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{17}{110}\right)\) \(e\left(\frac{199}{330}\right)\) \(e\left(\frac{49}{165}\right)\) \(e\left(\frac{49}{66}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{29}{110}\right)\) \(e\left(\frac{301}{330}\right)\) \(e\left(\frac{257}{330}\right)\)