from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1815, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([0,55,84]))
chi.galois_orbit()
[g,chi] = znchar(Mod(37,1815))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1815\) | |
| 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.x | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | 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 | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1815}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{220}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{203}{220}\right)\) | \(e\left(\frac{197}{220}\right)\) | \(e\left(\frac{69}{220}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{211}{220}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{21}{44}\right)\) |
| \(\chi_{1815}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{201}{220}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{197}{220}\right)\) | \(e\left(\frac{163}{220}\right)\) | \(e\left(\frac{171}{220}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{169}{220}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{31}{44}\right)\) |
| \(\chi_{1815}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{220}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{171}{220}\right)\) | \(e\left(\frac{89}{220}\right)\) | \(e\left(\frac{173}{220}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{207}{220}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{1}{44}\right)\) |
| \(\chi_{1815}(97,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{220}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{119}{220}\right)\) | \(e\left(\frac{161}{220}\right)\) | \(e\left(\frac{177}{220}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{63}{220}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{29}{44}\right)\) |
| \(\chi_{1815}(103,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{189}{220}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{113}{220}\right)\) | \(e\left(\frac{127}{220}\right)\) | \(e\left(\frac{59}{220}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{21}{220}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{39}{44}\right)\) |
| \(\chi_{1815}(157,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{220}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{127}{220}\right)\) | \(e\left(\frac{133}{220}\right)\) | \(e\left(\frac{41}{220}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{119}{220}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{1}{44}\right)\) |
| \(\chi_{1815}(163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{220}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{189}{220}\right)\) | \(e\left(\frac{191}{220}\right)\) | \(e\left(\frac{87}{220}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{113}{220}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{15}{44}\right)\) |
| \(\chi_{1815}(223,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{220}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{117}{220}\right)\) | \(e\left(\frac{3}{220}\right)\) | \(e\left(\frac{211}{220}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{49}{220}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{3}{44}\right)\) |
| \(\chi_{1815}(247,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{203}{220}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{211}{220}\right)\) | \(e\left(\frac{169}{220}\right)\) | \(e\left(\frac{153}{220}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{47}{220}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{37}{44}\right)\) |
| \(\chi_{1815}(262,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{207}{220}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{19}{220}\right)\) | \(e\left(\frac{181}{220}\right)\) | \(e\left(\frac{117}{220}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{23}{220}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{5}{44}\right)\) |
| \(\chi_{1815}(268,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{220}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{53}{220}\right)\) | \(e\left(\frac{7}{220}\right)\) | \(e\left(\frac{199}{220}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{41}{220}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{7}{44}\right)\) |
| \(\chi_{1815}(313,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{220}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{161}{220}\right)\) | \(e\left(\frac{179}{220}\right)\) | \(e\left(\frac{123}{220}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{137}{220}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{3}{44}\right)\) |
| \(\chi_{1815}(322,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{211}{220}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{47}{220}\right)\) | \(e\left(\frac{193}{220}\right)\) | \(e\left(\frac{81}{220}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{219}{220}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{17}{44}\right)\) |
| \(\chi_{1815}(328,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{217}{220}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{89}{220}\right)\) | \(e\left(\frac{211}{220}\right)\) | \(e\left(\frac{27}{220}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{73}{220}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{35}{44}\right)\) |
| \(\chi_{1815}(367,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{220}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{83}{220}\right)\) | \(e\left(\frac{177}{220}\right)\) | \(e\left(\frac{129}{220}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{31}{220}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{1}{44}\right)\) |
| \(\chi_{1815}(388,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{220}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{37}{220}\right)\) | \(e\left(\frac{63}{220}\right)\) | \(e\left(\frac{31}{220}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{149}{220}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{19}{44}\right)\) |
| \(\chi_{1815}(412,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{220}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{31}{220}\right)\) | \(e\left(\frac{29}{220}\right)\) | \(e\left(\frac{133}{220}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{107}{220}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{29}{44}\right)\) |
| \(\chi_{1815}(427,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{220}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{139}{220}\right)\) | \(e\left(\frac{201}{220}\right)\) | \(e\left(\frac{57}{220}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{203}{220}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{25}{44}\right)\) |
| \(\chi_{1815}(433,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{220}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{213}{220}\right)\) | \(e\left(\frac{107}{220}\right)\) | \(e\left(\frac{119}{220}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{61}{220}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{19}{44}\right)\) |
| \(\chi_{1815}(478,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{220}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{201}{220}\right)\) | \(e\left(\frac{39}{220}\right)\) | \(e\left(\frac{103}{220}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{197}{220}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{39}{44}\right)\) |
| \(\chi_{1815}(532,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{220}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{23}{220}\right)\) | \(e\left(\frac{57}{220}\right)\) | \(e\left(\frac{49}{220}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{51}{220}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{13}{44}\right)\) |
| \(\chi_{1815}(553,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{220}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{177}{220}\right)\) | \(e\left(\frac{123}{220}\right)\) | \(e\left(\frac{71}{220}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{29}{220}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{35}{44}\right)\) |
| \(\chi_{1815}(577,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{183}{220}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{71}{220}\right)\) | \(e\left(\frac{109}{220}\right)\) | \(e\left(\frac{113}{220}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{167}{220}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{21}{44}\right)\) |
| \(\chi_{1815}(592,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{147}{220}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{39}{220}\right)\) | \(e\left(\frac{1}{220}\right)\) | \(e\left(\frac{217}{220}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{163}{220}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{1}{44}\right)\) |
| \(\chi_{1815}(598,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{220}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{153}{220}\right)\) | \(e\left(\frac{207}{220}\right)\) | \(e\left(\frac{39}{220}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{81}{220}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{31}{44}\right)\) |
| \(\chi_{1815}(643,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{220}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{21}{220}\right)\) | \(e\left(\frac{119}{220}\right)\) | \(e\left(\frac{83}{220}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{37}{220}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{31}{44}\right)\) |
| \(\chi_{1815}(652,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{220}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{107}{220}\right)\) | \(e\left(\frac{93}{220}\right)\) | \(e\left(\frac{161}{220}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{199}{220}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{5}{44}\right)\) |
| \(\chi_{1815}(658,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{220}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{109}{220}\right)\) | \(e\left(\frac{31}{220}\right)\) | \(e\left(\frac{127}{220}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{213}{220}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{31}{44}\right)\) |
| \(\chi_{1815}(697,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{220}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{183}{220}\right)\) | \(e\left(\frac{157}{220}\right)\) | \(e\left(\frac{189}{220}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{71}{220}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{25}{44}\right)\) |
| \(\chi_{1815}(718,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{220}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{97}{220}\right)\) | \(e\left(\frac{183}{220}\right)\) | \(e\left(\frac{111}{220}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{129}{220}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{7}{44}\right)\) |
| \(\chi_{1815}(742,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{220}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{111}{220}\right)\) | \(e\left(\frac{189}{220}\right)\) | \(e\left(\frac{93}{220}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{7}{220}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{13}{44}\right)\) |