from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4600, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([0,110,99,210]))
chi.galois_orbit()
[g,chi] = znchar(Mod(37,4600))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4600\) | |
| Conductor: | \(4600\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(220\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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 | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4600}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{203}{220}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{91}{220}\right)\) | \(e\left(\frac{117}{220}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{169}{220}\right)\) | \(e\left(\frac{32}{55}\right)\) |
| \(\chi_{4600}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{169}{220}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{53}{220}\right)\) | \(e\left(\frac{131}{220}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{67}{220}\right)\) | \(e\left(\frac{41}{55}\right)\) |
| \(\chi_{4600}(237,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{220}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{31}{220}\right)\) | \(e\left(\frac{197}{220}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{89}{220}\right)\) | \(e\left(\frac{52}{55}\right)\) |
| \(\chi_{4600}(333,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{220}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{17}{220}\right)\) | \(e\left(\frac{179}{220}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{63}{220}\right)\) | \(e\left(\frac{9}{55}\right)\) |
| \(\chi_{4600}(373,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{220}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{129}{220}\right)\) | \(e\left(\frac{103}{220}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{51}{220}\right)\) | \(e\left(\frac{23}{55}\right)\) |
| \(\chi_{4600}(477,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{207}{220}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{199}{220}\right)\) | \(e\left(\frac{193}{220}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{181}{220}\right)\) | \(e\left(\frac{18}{55}\right)\) |
| \(\chi_{4600}(517,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{220}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{127}{220}\right)\) | \(e\left(\frac{69}{220}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{173}{220}\right)\) | \(e\left(\frac{9}{55}\right)\) |
| \(\chi_{4600}(573,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{177}{220}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{49}{220}\right)\) | \(e\left(\frac{63}{220}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{91}{220}\right)\) | \(e\left(\frac{13}{55}\right)\) |
| \(\chi_{4600}(613,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{220}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{81}{220}\right)\) | \(e\left(\frac{167}{220}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{119}{220}\right)\) | \(e\left(\frac{17}{55}\right)\) |
| \(\chi_{4600}(677,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{220}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{79}{220}\right)\) | \(e\left(\frac{133}{220}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{21}{220}\right)\) | \(e\left(\frac{3}{55}\right)\) |
| \(\chi_{4600}(733,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{220}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{117}{220}\right)\) | \(e\left(\frac{119}{220}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{123}{220}\right)\) | \(e\left(\frac{49}{55}\right)\) |
| \(\chi_{4600}(773,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{220}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{69}{220}\right)\) | \(e\left(\frac{183}{220}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{191}{220}\right)\) | \(e\left(\frac{43}{55}\right)\) |
| \(\chi_{4600}(797,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{220}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{103}{220}\right)\) | \(e\left(\frac{101}{220}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{97}{220}\right)\) | \(e\left(\frac{6}{55}\right)\) |
| \(\chi_{4600}(917,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{220}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{7}{220}\right)\) | \(e\left(\frac{9}{220}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{13}{220}\right)\) | \(e\left(\frac{49}{55}\right)\) |
| \(\chi_{4600}(973,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{220}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{9}{220}\right)\) | \(e\left(\frac{43}{220}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{111}{220}\right)\) | \(e\left(\frac{8}{55}\right)\) |
| \(\chi_{4600}(1077,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{220}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{219}{220}\right)\) | \(e\left(\frac{93}{220}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{61}{220}\right)\) | \(e\left(\frac{48}{55}\right)\) |
| \(\chi_{4600}(1213,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{220}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{1}{220}\right)\) | \(e\left(\frac{127}{220}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{159}{220}\right)\) | \(e\left(\frac{7}{55}\right)\) |
| \(\chi_{4600}(1253,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{220}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{193}{220}\right)\) | \(e\left(\frac{91}{220}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{107}{220}\right)\) | \(e\left(\frac{31}{55}\right)\) |
| \(\chi_{4600}(1397,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{220}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{23}{220}\right)\) | \(e\left(\frac{61}{220}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{137}{220}\right)\) | \(e\left(\frac{51}{55}\right)\) |
| \(\chi_{4600}(1413,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{220}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{101}{220}\right)\) | \(e\left(\frac{67}{220}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{219}{220}\right)\) | \(e\left(\frac{47}{55}\right)\) |
| \(\chi_{4600}(1437,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{220}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{171}{220}\right)\) | \(e\left(\frac{157}{220}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{129}{220}\right)\) | \(e\left(\frac{42}{55}\right)\) |
| \(\chi_{4600}(1477,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{220}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{19}{220}\right)\) | \(e\left(\frac{213}{220}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{161}{220}\right)\) | \(e\left(\frac{23}{55}\right)\) |
| \(\chi_{4600}(1533,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{201}{220}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{37}{220}\right)\) | \(e\left(\frac{79}{220}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{163}{220}\right)\) | \(e\left(\frac{39}{55}\right)\) |
| \(\chi_{4600}(1597,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{220}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{123}{220}\right)\) | \(e\left(\frac{1}{220}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{197}{220}\right)\) | \(e\left(\frac{36}{55}\right)\) |
| \(\chi_{4600}(1653,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{129}{220}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{73}{220}\right)\) | \(e\left(\frac{31}{220}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{167}{220}\right)\) | \(e\left(\frac{16}{55}\right)\) |
| \(\chi_{4600}(1677,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{220}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{159}{220}\right)\) | \(e\left(\frac{173}{220}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{201}{220}\right)\) | \(e\left(\frac{13}{55}\right)\) |
| \(\chi_{4600}(1717,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{220}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{147}{220}\right)\) | \(e\left(\frac{189}{220}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{53}{220}\right)\) | \(e\left(\frac{39}{55}\right)\) |
| \(\chi_{4600}(1813,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{220}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{21}{220}\right)\) | \(e\left(\frac{27}{220}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{39}{220}\right)\) | \(e\left(\frac{37}{55}\right)\) |
| \(\chi_{4600}(1837,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{220}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{51}{220}\right)\) | \(e\left(\frac{97}{220}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{189}{220}\right)\) | \(e\left(\frac{27}{55}\right)\) |
| \(\chi_{4600}(1877,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{220}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{179}{220}\right)\) | \(e\left(\frac{73}{220}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{81}{220}\right)\) | \(e\left(\frac{43}{55}\right)\) |
| \(\chi_{4600}(1997,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{220}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{43}{220}\right)\) | \(e\left(\frac{181}{220}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{17}{220}\right)\) | \(e\left(\frac{26}{55}\right)\) |