from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(605, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([55,2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,605))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(605\) | |
| Conductor: | \(605\) | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{605}(2,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{220}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{69}{220}\right)\) | \(e\left(\frac{171}{220}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{147}{220}\right)\) | \(e\left(\frac{63}{110}\right)\) |
| \(\chi_{605}(7,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{220}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{153}{220}\right)\) | \(e\left(\frac{207}{220}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{39}{220}\right)\) | \(e\left(\frac{1}{110}\right)\) |
| \(\chi_{605}(8,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{171}{220}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{207}{220}\right)\) | \(e\left(\frac{73}{220}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{1}{220}\right)\) | \(e\left(\frac{79}{110}\right)\) |
| \(\chi_{605}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{147}{220}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{39}{220}\right)\) | \(e\left(\frac{1}{220}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{217}{220}\right)\) | \(e\left(\frac{93}{110}\right)\) |
| \(\chi_{605}(17,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{153}{220}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{81}{220}\right)\) | \(e\left(\frac{19}{220}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{163}{220}\right)\) | \(e\left(\frac{7}{110}\right)\) |
| \(\chi_{605}(18,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{220}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{3}{220}\right)\) | \(e\left(\frac{17}{220}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{169}{220}\right)\) | \(e\left(\frac{41}{110}\right)\) |
| \(\chi_{605}(28,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{183}{220}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{71}{220}\right)\) | \(e\left(\frac{109}{220}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{113}{220}\right)\) | \(e\left(\frac{17}{110}\right)\) |
| \(\chi_{605}(52,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{220}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{177}{220}\right)\) | \(e\left(\frac{123}{220}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{71}{220}\right)\) | \(e\left(\frac{109}{110}\right)\) |
| \(\chi_{605}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{177}{220}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{29}{220}\right)\) | \(e\left(\frac{91}{220}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{167}{220}\right)\) | \(e\left(\frac{103}{110}\right)\) |
| \(\chi_{605}(62,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{220}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{173}{220}\right)\) | \(e\left(\frac{27}{220}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{139}{220}\right)\) | \(e\left(\frac{91}{110}\right)\) |
| \(\chi_{605}(63,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{220}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{87}{220}\right)\) | \(e\left(\frac{53}{220}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{61}{220}\right)\) | \(e\left(\frac{89}{110}\right)\) |
| \(\chi_{605}(68,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{220}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{219}{220}\right)\) | \(e\left(\frac{141}{220}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{17}{220}\right)\) | \(e\left(\frac{23}{110}\right)\) |
| \(\chi_{605}(72,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{220}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{141}{220}\right)\) | \(e\left(\frac{139}{220}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{23}{220}\right)\) | \(e\left(\frac{57}{110}\right)\) |
| \(\chi_{605}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{220}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{23}{220}\right)\) | \(e\left(\frac{57}{220}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{49}{220}\right)\) | \(e\left(\frac{21}{110}\right)\) |
| \(\chi_{605}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{220}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{131}{220}\right)\) | \(e\left(\frac{9}{220}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{193}{220}\right)\) | \(e\left(\frac{67}{110}\right)\) |
| \(\chi_{605}(107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{181}{220}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{57}{220}\right)\) | \(e\left(\frac{103}{220}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{131}{220}\right)\) | \(e\left(\frac{9}{110}\right)\) |
| \(\chi_{605}(117,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{169}{220}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{193}{220}\right)\) | \(e\left(\frac{67}{220}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{19}{220}\right)\) | \(e\left(\frac{71}{110}\right)\) |
| \(\chi_{605}(123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{220}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{179}{220}\right)\) | \(e\left(\frac{61}{220}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{37}{220}\right)\) | \(e\left(\frac{63}{110}\right)\) |
| \(\chi_{605}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{220}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{201}{220}\right)\) | \(e\left(\frac{39}{220}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{103}{220}\right)\) | \(e\left(\frac{107}{110}\right)\) |
| \(\chi_{605}(128,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{220}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{43}{220}\right)\) | \(e\left(\frac{97}{220}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{149}{220}\right)\) | \(e\left(\frac{1}{110}\right)\) |
| \(\chi_{605}(138,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{220}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{191}{220}\right)\) | \(e\left(\frac{129}{220}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{53}{220}\right)\) | \(e\left(\frac{7}{110}\right)\) |
| \(\chi_{605}(162,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{220}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{157}{220}\right)\) | \(e\left(\frac{83}{220}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{191}{220}\right)\) | \(e\left(\frac{19}{110}\right)\) |
| \(\chi_{605}(167,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{197}{220}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{169}{220}\right)\) | \(e\left(\frac{151}{220}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{207}{220}\right)\) | \(e\left(\frac{73}{110}\right)\) |
| \(\chi_{605}(172,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{220}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{213}{220}\right)\) | \(e\left(\frac{107}{220}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{119}{220}\right)\) | \(e\left(\frac{51}{110}\right)\) |
| \(\chi_{605}(173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{220}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{67}{220}\right)\) | \(e\left(\frac{13}{220}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{181}{220}\right)\) | \(e\left(\frac{109}{110}\right)\) |
| \(\chi_{605}(178,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{220}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{139}{220}\right)\) | \(e\left(\frac{201}{220}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{57}{220}\right)\) | \(e\left(\frac{103}{110}\right)\) |
| \(\chi_{605}(182,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{220}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{41}{220}\right)\) | \(e\left(\frac{159}{220}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{183}{220}\right)\) | \(e\left(\frac{47}{110}\right)\) |
| \(\chi_{605}(183,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{220}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{63}{220}\right)\) | \(e\left(\frac{137}{220}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{29}{220}\right)\) | \(e\left(\frac{91}{110}\right)\) |
| \(\chi_{605}(193,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{220}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{31}{220}\right)\) | \(e\left(\frac{29}{220}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{133}{220}\right)\) | \(e\left(\frac{57}{110}\right)\) |
| \(\chi_{605}(217,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{220}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{37}{220}\right)\) | \(e\left(\frac{63}{220}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{31}{220}\right)\) | \(e\left(\frac{29}{110}\right)\) |
| \(\chi_{605}(222,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{220}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{129}{220}\right)\) | \(e\left(\frac{71}{220}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{7}{220}\right)\) | \(e\left(\frac{3}{110}\right)\) |