from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1013, base_ring=CyclotomicField(506))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(9,1013))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1013\) | |
| Conductor: | \(1013\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(506\) | 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: | $\Q(\zeta_{253})$ |
| Fixed field: | Number field defined by a degree 506 polynomial (not computed) |
First 31 of 220 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1013}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{1}{506}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{251}{506}\right)\) | \(e\left(\frac{182}{253}\right)\) | \(e\left(\frac{315}{506}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{1}{253}\right)\) | \(e\left(\frac{54}{253}\right)\) | \(e\left(\frac{4}{23}\right)\) |
| \(\chi_{1013}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{415}{506}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{435}{506}\right)\) | \(e\left(\frac{136}{253}\right)\) | \(e\left(\frac{177}{506}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{162}{253}\right)\) | \(e\left(\frac{146}{253}\right)\) | \(e\left(\frac{4}{23}\right)\) |
| \(\chi_{1013}(15,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{379}{506}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{1}{506}\right)\) | \(e\left(\frac{162}{253}\right)\) | \(e\left(\frac{475}{506}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{126}{253}\right)\) | \(e\left(\frac{226}{253}\right)\) | \(e\left(\frac{21}{23}\right)\) |
| \(\chi_{1013}(21,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{411}{506}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{443}{506}\right)\) | \(e\left(\frac{167}{253}\right)\) | \(e\left(\frac{435}{506}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{158}{253}\right)\) | \(e\left(\frac{183}{253}\right)\) | \(e\left(\frac{11}{23}\right)\) |
| \(\chi_{1013}(24,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{39}{506}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{175}{506}\right)\) | \(e\left(\frac{14}{253}\right)\) | \(e\left(\frac{141}{506}\right)\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{39}{253}\right)\) | \(e\left(\frac{82}{253}\right)\) | \(e\left(\frac{18}{23}\right)\) |
| \(\chi_{1013}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{251}{506}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{257}{506}\right)\) | \(e\left(\frac{142}{253}\right)\) | \(e\left(\frac{129}{506}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{251}{253}\right)\) | \(e\left(\frac{145}{253}\right)\) | \(e\left(\frac{15}{23}\right)\) |
| \(\chi_{1013}(35,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{283}{506}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{193}{506}\right)\) | \(e\left(\frac{147}{253}\right)\) | \(e\left(\frac{89}{506}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{30}{253}\right)\) | \(e\left(\frac{102}{253}\right)\) | \(e\left(\frac{5}{23}\right)\) |
| \(\chi_{1013}(40,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{417}{506}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{431}{506}\right)\) | \(e\left(\frac{247}{253}\right)\) | \(e\left(\frac{301}{506}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{164}{253}\right)\) | \(e\left(\frac{1}{253}\right)\) | \(e\left(\frac{12}{23}\right)\) |
| \(\chi_{1013}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{193}{506}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{373}{506}\right)\) | \(e\left(\frac{212}{253}\right)\) | \(e\left(\frac{75}{506}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{193}{253}\right)\) | \(e\left(\frac{49}{253}\right)\) | \(e\left(\frac{13}{23}\right)\) |
| \(\chi_{1013}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{315}{506}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{129}{506}\right)\) | \(e\left(\frac{152}{253}\right)\) | \(e\left(\frac{49}{506}\right)\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{62}{253}\right)\) | \(e\left(\frac{59}{253}\right)\) | \(e\left(\frac{18}{23}\right)\) |
| \(\chi_{1013}(51,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{175}{506}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{409}{506}\right)\) | \(e\left(\frac{225}{253}\right)\) | \(e\left(\frac{477}{506}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{175}{253}\right)\) | \(e\left(\frac{89}{253}\right)\) | \(e\left(\frac{10}{23}\right)\) |
| \(\chi_{1013}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{203}{506}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{353}{506}\right)\) | \(e\left(\frac{8}{253}\right)\) | \(e\left(\frac{189}{506}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{203}{253}\right)\) | \(e\left(\frac{83}{253}\right)\) | \(e\left(\frac{7}{23}\right)\) |
| \(\chi_{1013}(54,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{183}{506}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{393}{506}\right)\) | \(e\left(\frac{163}{253}\right)\) | \(e\left(\frac{467}{506}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{183}{253}\right)\) | \(e\left(\frac{15}{253}\right)\) | \(e\left(\frac{19}{23}\right)\) |
| \(\chi_{1013}(56,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{449}{506}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{367}{506}\right)\) | \(e\left(\frac{252}{253}\right)\) | \(e\left(\frac{261}{506}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{196}{253}\right)\) | \(e\left(\frac{211}{253}\right)\) | \(e\left(\frac{2}{23}\right)\) |
| \(\chi_{1013}(66,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{479}{506}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{307}{506}\right)\) | \(e\left(\frac{146}{253}\right)\) | \(e\left(\frac{97}{506}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{226}{253}\right)\) | \(e\left(\frac{60}{253}\right)\) | \(e\left(\frac{7}{23}\right)\) |
| \(\chi_{1013}(71,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{399}{506}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{467}{506}\right)\) | \(e\left(\frac{7}{253}\right)\) | \(e\left(\frac{197}{506}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{146}{253}\right)\) | \(e\left(\frac{41}{253}\right)\) | \(e\left(\frac{9}{23}\right)\) |
| \(\chi_{1013}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{421}{506}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{423}{506}\right)\) | \(e\left(\frac{216}{253}\right)\) | \(e\left(\frac{43}{506}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{168}{253}\right)\) | \(e\left(\frac{217}{253}\right)\) | \(e\left(\frac{5}{23}\right)\) |
| \(\chi_{1013}(74,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{13}{506}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{227}{506}\right)\) | \(e\left(\frac{89}{253}\right)\) | \(e\left(\frac{47}{506}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{13}{253}\right)\) | \(e\left(\frac{196}{253}\right)\) | \(e\left(\frac{6}{23}\right)\) |
| \(\chi_{1013}(76,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{89}{506}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{75}{506}\right)\) | \(e\left(\frac{6}{253}\right)\) | \(e\left(\frac{205}{506}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{89}{253}\right)\) | \(e\left(\frac{252}{253}\right)\) | \(e\left(\frac{11}{23}\right)\) |
| \(\chi_{1013}(78,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{91}{506}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{71}{506}\right)\) | \(e\left(\frac{117}{253}\right)\) | \(e\left(\frac{329}{506}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{91}{253}\right)\) | \(e\left(\frac{107}{253}\right)\) | \(e\left(\frac{19}{23}\right)\) |
| \(\chi_{1013}(79,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{177}{506}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{405}{506}\right)\) | \(e\left(\frac{83}{253}\right)\) | \(e\left(\frac{95}{506}\right)\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{177}{253}\right)\) | \(e\left(\frac{197}{253}\right)\) | \(e\left(\frac{18}{23}\right)\) |
| \(\chi_{1013}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{47}{506}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{159}{506}\right)\) | \(e\left(\frac{205}{253}\right)\) | \(e\left(\frac{131}{506}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{47}{253}\right)\) | \(e\left(\frac{8}{253}\right)\) | \(e\left(\frac{4}{23}\right)\) |
| \(\chi_{1013}(87,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{405}{506}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{455}{506}\right)\) | \(e\left(\frac{87}{253}\right)\) | \(e\left(\frac{63}{506}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{152}{253}\right)\) | \(e\left(\frac{112}{253}\right)\) | \(e\left(\frac{10}{23}\right)\) |
| \(\chi_{1013}(93,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{263}{506}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{233}{506}\right)\) | \(e\left(\frac{49}{253}\right)\) | \(e\left(\frac{367}{506}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{10}{253}\right)\) | \(e\left(\frac{34}{253}\right)\) | \(e\left(\frac{17}{23}\right)\) |
| \(\chi_{1013}(110,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{351}{506}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{57}{506}\right)\) | \(e\left(\frac{126}{253}\right)\) | \(e\left(\frac{257}{506}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{98}{253}\right)\) | \(e\left(\frac{232}{253}\right)\) | \(e\left(\frac{1}{23}\right)\) |
| \(\chi_{1013}(119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{79}{506}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{95}{506}\right)\) | \(e\left(\frac{210}{253}\right)\) | \(e\left(\frac{91}{506}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{79}{253}\right)\) | \(e\left(\frac{218}{253}\right)\) | \(e\left(\frac{17}{23}\right)\) |
| \(\chi_{1013}(123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{113}{506}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{27}{506}\right)\) | \(e\left(\frac{73}{253}\right)\) | \(e\left(\frac{175}{506}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{113}{253}\right)\) | \(e\left(\frac{30}{253}\right)\) | \(e\left(\frac{15}{23}\right)\) |
| \(\chi_{1013}(126,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{87}{506}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{79}{506}\right)\) | \(e\left(\frac{148}{253}\right)\) | \(e\left(\frac{81}{506}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{87}{253}\right)\) | \(e\left(\frac{144}{253}\right)\) | \(e\left(\frac{3}{23}\right)\) |
| \(\chi_{1013}(130,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{469}{506}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{327}{506}\right)\) | \(e\left(\frac{97}{253}\right)\) | \(e\left(\frac{489}{506}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{216}{253}\right)\) | \(e\left(\frac{26}{253}\right)\) | \(e\left(\frac{13}{23}\right)\) |
| \(\chi_{1013}(136,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{213}{506}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{333}{506}\right)\) | \(e\left(\frac{57}{253}\right)\) | \(e\left(\frac{303}{506}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{213}{253}\right)\) | \(e\left(\frac{117}{253}\right)\) | \(e\left(\frac{1}{23}\right)\) |
| \(\chi_{1013}(138,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{281}{506}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{197}{506}\right)\) | \(e\left(\frac{36}{253}\right)\) | \(e\left(\frac{471}{506}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{28}{253}\right)\) | \(e\left(\frac{247}{253}\right)\) | \(e\left(\frac{20}{23}\right)\) |