Properties

Label 8470.dg
Modulus $8470$
Conductor $605$
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([55,0,122]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(57,8470))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(8470\)
Conductor: \(605\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(220\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from 605.w
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 \(-1\) \(1\) \(3\) \(9\) \(13\) \(17\) \(19\) \(23\) \(27\) \(29\) \(31\) \(37\)
\(\chi_{8470}(57,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{167}{220}\right)\) \(e\left(\frac{93}{220}\right)\) \(e\left(\frac{29}{55}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{51}{55}\right)\) \(e\left(\frac{38}{55}\right)\) \(e\left(\frac{119}{220}\right)\)
\(\chi_{8470}(127,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{103}{220}\right)\) \(e\left(\frac{197}{220}\right)\) \(e\left(\frac{36}{55}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{14}{55}\right)\) \(e\left(\frac{32}{55}\right)\) \(e\left(\frac{51}{220}\right)\)
\(\chi_{8470}(183,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{29}{220}\right)\) \(e\left(\frac{111}{220}\right)\) \(e\left(\frac{8}{55}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{52}{55}\right)\) \(e\left(\frac{1}{55}\right)\) \(e\left(\frac{213}{220}\right)\)
\(\chi_{8470}(337,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{199}{220}\right)\) \(e\left(\frac{41}{220}\right)\) \(e\left(\frac{53}{55}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{42}{55}\right)\) \(e\left(\frac{41}{55}\right)\) \(e\left(\frac{43}{220}\right)\)
\(\chi_{8470}(393,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{201}{220}\right)\) \(e\left(\frac{79}{220}\right)\) \(e\left(\frac{27}{55}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{38}{55}\right)\) \(e\left(\frac{24}{55}\right)\) \(e\left(\frac{217}{220}\right)\)
\(\chi_{8470}(547,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{171}{220}\right)\) \(e\left(\frac{169}{220}\right)\) \(e\left(\frac{32}{55}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{43}{55}\right)\) \(e\left(\frac{4}{55}\right)\) \(e\left(\frac{27}{220}\right)\)
\(\chi_{8470}(673,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{17}{220}\right)\) \(e\left(\frac{103}{220}\right)\) \(e\left(\frac{54}{55}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{21}{55}\right)\) \(e\left(\frac{48}{55}\right)\) \(e\left(\frac{49}{220}\right)\)
\(\chi_{8470}(743,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{53}{220}\right)\) \(e\left(\frac{127}{220}\right)\) \(e\left(\frac{26}{55}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{4}{55}\right)\) \(e\left(\frac{17}{55}\right)\) \(e\left(\frac{101}{220}\right)\)
\(\chi_{8470}(827,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{7}{220}\right)\) \(e\left(\frac{133}{220}\right)\) \(e\left(\frac{19}{55}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{41}{55}\right)\) \(e\left(\frac{23}{55}\right)\) \(e\left(\frac{59}{220}\right)\)
\(\chi_{8470}(897,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{123}{220}\right)\) \(e\left(\frac{137}{220}\right)\) \(e\left(\frac{51}{55}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{29}{55}\right)\) \(e\left(\frac{27}{55}\right)\) \(e\left(\frac{31}{220}\right)\)
\(\chi_{8470}(953,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{109}{220}\right)\) \(e\left(\frac{91}{220}\right)\) \(e\left(\frac{13}{55}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{2}{55}\right)\) \(e\left(\frac{36}{55}\right)\) \(e\left(\frac{133}{220}\right)\)
\(\chi_{8470}(1107,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{59}{220}\right)\) \(e\left(\frac{21}{220}\right)\) \(e\left(\frac{3}{55}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{47}{55}\right)\) \(e\left(\frac{21}{55}\right)\) \(e\left(\frac{183}{220}\right)\)
\(\chi_{8470}(1163,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{161}{220}\right)\) \(e\left(\frac{199}{220}\right)\) \(e\left(\frac{52}{55}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{8}{55}\right)\) \(e\left(\frac{34}{55}\right)\) \(e\left(\frac{37}{220}\right)\)
\(\chi_{8470}(1317,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{131}{220}\right)\) \(e\left(\frac{69}{220}\right)\) \(e\left(\frac{2}{55}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{13}{55}\right)\) \(e\left(\frac{14}{55}\right)\) \(e\left(\frac{67}{220}\right)\)
\(\chi_{8470}(1513,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{73}{220}\right)\) \(e\left(\frac{67}{220}\right)\) \(e\left(\frac{41}{55}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{19}{55}\right)\) \(e\left(\frac{12}{55}\right)\) \(e\left(\frac{81}{220}\right)\)
\(\chi_{8470}(1597,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{67}{220}\right)\) \(e\left(\frac{173}{220}\right)\) \(e\left(\frac{9}{55}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{31}{55}\right)\) \(e\left(\frac{8}{55}\right)\) \(e\left(\frac{219}{220}\right)\)
\(\chi_{8470}(1723,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{189}{220}\right)\) \(e\left(\frac{71}{220}\right)\) \(e\left(\frac{18}{55}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{7}{55}\right)\) \(e\left(\frac{16}{55}\right)\) \(e\left(\frac{53}{220}\right)\)
\(\chi_{8470}(1877,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{139}{220}\right)\) \(e\left(\frac{1}{220}\right)\) \(e\left(\frac{8}{55}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{52}{55}\right)\) \(e\left(\frac{1}{55}\right)\) \(e\left(\frac{103}{220}\right)\)
\(\chi_{8470}(2087,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{91}{220}\right)\) \(e\left(\frac{189}{220}\right)\) \(e\left(\frac{27}{55}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{38}{55}\right)\) \(e\left(\frac{24}{55}\right)\) \(e\left(\frac{107}{220}\right)\)
\(\chi_{8470}(2213,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{137}{220}\right)\) \(e\left(\frac{183}{220}\right)\) \(e\left(\frac{34}{55}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{1}{55}\right)\) \(e\left(\frac{18}{55}\right)\) \(e\left(\frac{149}{220}\right)\)
\(\chi_{8470}(2283,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{93}{220}\right)\) \(e\left(\frac{7}{220}\right)\) \(e\left(\frac{1}{55}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{34}{55}\right)\) \(e\left(\frac{7}{55}\right)\) \(e\left(\frac{61}{220}\right)\)
\(\chi_{8470}(2367,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{127}{220}\right)\) \(e\left(\frac{213}{220}\right)\) \(e\left(\frac{54}{55}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{21}{55}\right)\) \(e\left(\frac{48}{55}\right)\) \(e\left(\frac{159}{220}\right)\)
\(\chi_{8470}(2437,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{163}{220}\right)\) \(e\left(\frac{17}{220}\right)\) \(e\left(\frac{26}{55}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{4}{55}\right)\) \(e\left(\frac{17}{55}\right)\) \(e\left(\frac{211}{220}\right)\)
\(\chi_{8470}(2493,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{49}{220}\right)\) \(e\left(\frac{51}{220}\right)\) \(e\left(\frac{23}{55}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{12}{55}\right)\) \(e\left(\frac{51}{55}\right)\) \(e\left(\frac{193}{220}\right)\)
\(\chi_{8470}(2647,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{219}{220}\right)\) \(e\left(\frac{201}{220}\right)\) \(e\left(\frac{13}{55}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{2}{55}\right)\) \(e\left(\frac{36}{55}\right)\) \(e\left(\frac{23}{220}\right)\)
\(\chi_{8470}(2703,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{81}{220}\right)\) \(e\left(\frac{219}{220}\right)\) \(e\left(\frac{47}{55}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{3}{55}\right)\) \(e\left(\frac{54}{55}\right)\) \(e\left(\frac{117}{220}\right)\)
\(\chi_{8470}(2857,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{51}{220}\right)\) \(e\left(\frac{89}{220}\right)\) \(e\left(\frac{52}{55}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{8}{55}\right)\) \(e\left(\frac{34}{55}\right)\) \(e\left(\frac{147}{220}\right)\)
\(\chi_{8470}(2983,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{197}{220}\right)\) \(e\left(\frac{3}{220}\right)\) \(e\left(\frac{24}{55}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{46}{55}\right)\) \(e\left(\frac{3}{55}\right)\) \(e\left(\frac{89}{220}\right)\)
\(\chi_{8470}(3053,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{113}{220}\right)\) \(e\left(\frac{167}{220}\right)\) \(e\left(\frac{16}{55}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{49}{55}\right)\) \(e\left(\frac{2}{55}\right)\) \(e\left(\frac{41}{220}\right)\)
\(\chi_{8470}(3207,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{183}{220}\right)\) \(e\left(\frac{177}{220}\right)\) \(e\left(\frac{41}{55}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{19}{55}\right)\) \(e\left(\frac{12}{55}\right)\) \(e\left(\frac{191}{220}\right)\)
\(\chi_{8470}(3263,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{129}{220}\right)\) \(e\left(\frac{31}{220}\right)\) \(e\left(\frac{28}{55}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{17}{55}\right)\) \(e\left(\frac{31}{55}\right)\) \(e\left(\frac{113}{220}\right)\)