from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8470, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([55,0,122]))
chi.galois_orbit()
[g,chi] = znchar(Mod(57,8470))
order = charorder(g,chi)
[ 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 | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8470}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{167}{220}\right)\) | \(e\left(\frac{93}{220}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{119}{220}\right)\) |
| \(\chi_{8470}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{103}{220}\right)\) | \(e\left(\frac{197}{220}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{51}{220}\right)\) |
| \(\chi_{8470}(183,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{29}{220}\right)\) | \(e\left(\frac{111}{220}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{213}{220}\right)\) |
| \(\chi_{8470}(337,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{199}{220}\right)\) | \(e\left(\frac{41}{220}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{43}{220}\right)\) |
| \(\chi_{8470}(393,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{201}{220}\right)\) | \(e\left(\frac{79}{220}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{217}{220}\right)\) |
| \(\chi_{8470}(547,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{171}{220}\right)\) | \(e\left(\frac{169}{220}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{27}{220}\right)\) |
| \(\chi_{8470}(673,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{17}{220}\right)\) | \(e\left(\frac{103}{220}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{49}{220}\right)\) |
| \(\chi_{8470}(743,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{53}{220}\right)\) | \(e\left(\frac{127}{220}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{101}{220}\right)\) |
| \(\chi_{8470}(827,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{220}\right)\) | \(e\left(\frac{133}{220}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{59}{220}\right)\) |
| \(\chi_{8470}(897,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{123}{220}\right)\) | \(e\left(\frac{137}{220}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{31}{220}\right)\) |
| \(\chi_{8470}(953,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{109}{220}\right)\) | \(e\left(\frac{91}{220}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{133}{220}\right)\) |
| \(\chi_{8470}(1107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{59}{220}\right)\) | \(e\left(\frac{21}{220}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{183}{220}\right)\) |
| \(\chi_{8470}(1163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{161}{220}\right)\) | \(e\left(\frac{199}{220}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{37}{220}\right)\) |
| \(\chi_{8470}(1317,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{131}{220}\right)\) | \(e\left(\frac{69}{220}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{67}{220}\right)\) |
| \(\chi_{8470}(1513,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{73}{220}\right)\) | \(e\left(\frac{67}{220}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{81}{220}\right)\) |
| \(\chi_{8470}(1597,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{67}{220}\right)\) | \(e\left(\frac{173}{220}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{219}{220}\right)\) |
| \(\chi_{8470}(1723,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{189}{220}\right)\) | \(e\left(\frac{71}{220}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{53}{220}\right)\) |
| \(\chi_{8470}(1877,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{139}{220}\right)\) | \(e\left(\frac{1}{220}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{103}{220}\right)\) |
| \(\chi_{8470}(2087,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{91}{220}\right)\) | \(e\left(\frac{189}{220}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{107}{220}\right)\) |
| \(\chi_{8470}(2213,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{137}{220}\right)\) | \(e\left(\frac{183}{220}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{149}{220}\right)\) |
| \(\chi_{8470}(2283,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{93}{220}\right)\) | \(e\left(\frac{7}{220}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{61}{220}\right)\) |
| \(\chi_{8470}(2367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{127}{220}\right)\) | \(e\left(\frac{213}{220}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{159}{220}\right)\) |
| \(\chi_{8470}(2437,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{163}{220}\right)\) | \(e\left(\frac{17}{220}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{211}{220}\right)\) |
| \(\chi_{8470}(2493,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{49}{220}\right)\) | \(e\left(\frac{51}{220}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{193}{220}\right)\) |
| \(\chi_{8470}(2647,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{219}{220}\right)\) | \(e\left(\frac{201}{220}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{23}{220}\right)\) |
| \(\chi_{8470}(2703,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{81}{220}\right)\) | \(e\left(\frac{219}{220}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{117}{220}\right)\) |
| \(\chi_{8470}(2857,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{51}{220}\right)\) | \(e\left(\frac{89}{220}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{147}{220}\right)\) |
| \(\chi_{8470}(2983,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{197}{220}\right)\) | \(e\left(\frac{3}{220}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{89}{220}\right)\) |
| \(\chi_{8470}(3053,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{113}{220}\right)\) | \(e\left(\frac{167}{220}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{41}{220}\right)\) |
| \(\chi_{8470}(3207,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{183}{220}\right)\) | \(e\left(\frac{177}{220}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{191}{220}\right)\) |
| \(\chi_{8470}(3263,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{129}{220}\right)\) | \(e\left(\frac{31}{220}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{113}{220}\right)\) |