Properties

Label 8470.df
Modulus $8470$
Conductor $4235$
Order $220$
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(220))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([55,110,72]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(97,8470))
 
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: \(220\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from 4235.dd
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: $\Q(\zeta_{220})$
Fixed field: Number field defined by a degree 220 polynomial (not computed)

First 31 of 80 characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(9\) \(13\) \(17\) \(19\) \(23\) \(27\) \(29\) \(31\) \(37\)
\(\chi_{8470}(97,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{67}{220}\right)\) \(e\left(\frac{173}{220}\right)\) \(e\left(\frac{9}{55}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{7}{110}\right)\) \(e\left(\frac{71}{110}\right)\) \(e\left(\frac{219}{220}\right)\)
\(\chi_{8470}(223,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{101}{220}\right)\) \(e\left(\frac{159}{220}\right)\) \(e\left(\frac{7}{55}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{91}{110}\right)\) \(e\left(\frac{43}{110}\right)\) \(e\left(\frac{97}{220}\right)\)
\(\chi_{8470}(377,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{131}{220}\right)\) \(e\left(\frac{69}{220}\right)\) \(e\left(\frac{2}{55}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{81}{110}\right)\) \(e\left(\frac{83}{110}\right)\) \(e\left(\frac{67}{220}\right)\)
\(\chi_{8470}(433,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{9}{220}\right)\) \(e\left(\frac{171}{220}\right)\) \(e\left(\frac{48}{55}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{19}{110}\right)\) \(e\left(\frac{67}{110}\right)\) \(e\left(\frac{13}{220}\right)\)
\(\chi_{8470}(587,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{59}{220}\right)\) \(e\left(\frac{21}{220}\right)\) \(e\left(\frac{3}{55}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{39}{110}\right)\) \(e\left(\frac{97}{110}\right)\) \(e\left(\frac{183}{220}\right)\)
\(\chi_{8470}(643,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{193}{220}\right)\) \(e\left(\frac{147}{220}\right)\) \(e\left(\frac{21}{55}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{53}{110}\right)\) \(e\left(\frac{19}{110}\right)\) \(e\left(\frac{181}{220}\right)\)
\(\chi_{8470}(713,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{217}{220}\right)\) \(e\left(\frac{163}{220}\right)\) \(e\left(\frac{39}{55}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{67}{110}\right)\) \(e\left(\frac{51}{110}\right)\) \(e\left(\frac{69}{220}\right)\)
\(\chi_{8470}(797,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{123}{220}\right)\) \(e\left(\frac{137}{220}\right)\) \(e\left(\frac{51}{55}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{3}{110}\right)\) \(e\left(\frac{109}{110}\right)\) \(e\left(\frac{31}{220}\right)\)
\(\chi_{8470}(867,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{7}{220}\right)\) \(e\left(\frac{133}{220}\right)\) \(e\left(\frac{19}{55}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{27}{110}\right)\) \(e\left(\frac{101}{110}\right)\) \(e\left(\frac{59}{220}\right)\)
\(\chi_{8470}(993,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{141}{220}\right)\) \(e\left(\frac{39}{220}\right)\) \(e\left(\frac{37}{55}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{41}{110}\right)\) \(e\left(\frac{23}{110}\right)\) \(e\left(\frac{57}{220}\right)\)
\(\chi_{8470}(1147,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{171}{220}\right)\) \(e\left(\frac{169}{220}\right)\) \(e\left(\frac{32}{55}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{31}{110}\right)\) \(e\left(\frac{63}{110}\right)\) \(e\left(\frac{27}{220}\right)\)
\(\chi_{8470}(1203,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{149}{220}\right)\) \(e\left(\frac{191}{220}\right)\) \(e\left(\frac{43}{55}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{9}{110}\right)\) \(e\left(\frac{107}{110}\right)\) \(e\left(\frac{93}{220}\right)\)
\(\chi_{8470}(1357,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{199}{220}\right)\) \(e\left(\frac{41}{220}\right)\) \(e\left(\frac{53}{55}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{29}{110}\right)\) \(e\left(\frac{27}{110}\right)\) \(e\left(\frac{43}{220}\right)\)
\(\chi_{8470}(1413,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{173}{220}\right)\) \(e\left(\frac{207}{220}\right)\) \(e\left(\frac{6}{55}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{23}{110}\right)\) \(e\left(\frac{29}{110}\right)\) \(e\left(\frac{201}{220}\right)\)
\(\chi_{8470}(1483,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{157}{220}\right)\) \(e\left(\frac{123}{220}\right)\) \(e\left(\frac{49}{55}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{87}{110}\right)\) \(e\left(\frac{81}{110}\right)\) \(e\left(\frac{129}{220}\right)\)
\(\chi_{8470}(1567,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{103}{220}\right)\) \(e\left(\frac{197}{220}\right)\) \(e\left(\frac{36}{55}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{83}{110}\right)\) \(e\left(\frac{9}{110}\right)\) \(e\left(\frac{51}{220}\right)\)
\(\chi_{8470}(1637,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{167}{220}\right)\) \(e\left(\frac{93}{220}\right)\) \(e\left(\frac{29}{55}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{47}{110}\right)\) \(e\left(\frac{21}{110}\right)\) \(e\left(\frac{119}{220}\right)\)
\(\chi_{8470}(1763,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{181}{220}\right)\) \(e\left(\frac{139}{220}\right)\) \(e\left(\frac{12}{55}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{101}{110}\right)\) \(e\left(\frac{3}{110}\right)\) \(e\left(\frac{17}{220}\right)\)
\(\chi_{8470}(1917,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{211}{220}\right)\) \(e\left(\frac{49}{220}\right)\) \(e\left(\frac{7}{55}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{91}{110}\right)\) \(e\left(\frac{43}{110}\right)\) \(e\left(\frac{207}{220}\right)\)
\(\chi_{8470}(1973,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{69}{220}\right)\) \(e\left(\frac{211}{220}\right)\) \(e\left(\frac{38}{55}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{109}{110}\right)\) \(e\left(\frac{37}{110}\right)\) \(e\left(\frac{173}{220}\right)\)
\(\chi_{8470}(2127,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{119}{220}\right)\) \(e\left(\frac{61}{220}\right)\) \(e\left(\frac{48}{55}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{19}{110}\right)\) \(e\left(\frac{67}{110}\right)\) \(e\left(\frac{123}{220}\right)\)
\(\chi_{8470}(2183,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{153}{220}\right)\) \(e\left(\frac{47}{220}\right)\) \(e\left(\frac{46}{55}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{103}{110}\right)\) \(e\left(\frac{39}{110}\right)\) \(e\left(\frac{1}{220}\right)\)
\(\chi_{8470}(2253,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{97}{220}\right)\) \(e\left(\frac{83}{220}\right)\) \(e\left(\frac{4}{55}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{107}{110}\right)\) \(e\left(\frac{1}{110}\right)\) \(e\left(\frac{189}{220}\right)\)
\(\chi_{8470}(2337,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{83}{220}\right)\) \(e\left(\frac{37}{220}\right)\) \(e\left(\frac{21}{55}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{53}{110}\right)\) \(e\left(\frac{19}{110}\right)\) \(e\left(\frac{71}{220}\right)\)
\(\chi_{8470}(2407,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{107}{220}\right)\) \(e\left(\frac{53}{220}\right)\) \(e\left(\frac{39}{55}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{67}{110}\right)\) \(e\left(\frac{51}{110}\right)\) \(e\left(\frac{179}{220}\right)\)
\(\chi_{8470}(2533,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{220}\right)\) \(e\left(\frac{19}{220}\right)\) \(e\left(\frac{42}{55}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{51}{110}\right)\) \(e\left(\frac{93}{110}\right)\) \(e\left(\frac{197}{220}\right)\)
\(\chi_{8470}(2687,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{31}{220}\right)\) \(e\left(\frac{149}{220}\right)\) \(e\left(\frac{37}{55}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{41}{110}\right)\) \(e\left(\frac{23}{110}\right)\) \(e\left(\frac{167}{220}\right)\)
\(\chi_{8470}(2897,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{39}{220}\right)\) \(e\left(\frac{81}{220}\right)\) \(e\left(\frac{43}{55}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{9}{110}\right)\) \(e\left(\frac{107}{110}\right)\) \(e\left(\frac{203}{220}\right)\)
\(\chi_{8470}(2953,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{133}{220}\right)\) \(e\left(\frac{107}{220}\right)\) \(e\left(\frac{31}{55}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{73}{110}\right)\) \(e\left(\frac{49}{110}\right)\) \(e\left(\frac{21}{220}\right)\)
\(\chi_{8470}(3023,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{37}{220}\right)\) \(e\left(\frac{43}{220}\right)\) \(e\left(\frac{14}{55}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{17}{110}\right)\) \(e\left(\frac{31}{110}\right)\) \(e\left(\frac{29}{220}\right)\)
\(\chi_{8470}(3107,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{63}{220}\right)\) \(e\left(\frac{97}{220}\right)\) \(e\left(\frac{6}{55}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{23}{110}\right)\) \(e\left(\frac{29}{110}\right)\) \(e\left(\frac{91}{220}\right)\)