from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9802, base_ring=CyclotomicField(1092))
M = H._module
chi = DirichletCharacter(H, M([1057,117]))
chi.galois_orbit()
[g,chi] = znchar(Mod(37,9802))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(9802\) | |
| Conductor: | \(4901\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1092\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 4901.db | 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: | $\Q(\zeta_{1092})$ |
| Fixed field: | Number field defined by a degree 1092 polynomial (not computed) |
First 31 of 288 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{9802}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{613}{1092}\right)\) | \(e\left(\frac{25}{364}\right)\) | \(e\left(\frac{935}{1092}\right)\) | \(e\left(\frac{67}{546}\right)\) | \(e\left(\frac{103}{273}\right)\) | \(e\left(\frac{172}{273}\right)\) | \(e\left(\frac{89}{156}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{38}{91}\right)\) | \(e\left(\frac{41}{42}\right)\) |
| \(\chi_{9802}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{965}{1092}\right)\) | \(e\left(\frac{265}{364}\right)\) | \(e\left(\frac{811}{1092}\right)\) | \(e\left(\frac{419}{546}\right)\) | \(e\left(\frac{200}{273}\right)\) | \(e\left(\frac{167}{273}\right)\) | \(e\left(\frac{49}{156}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{57}{91}\right)\) | \(e\left(\frac{37}{42}\right)\) |
| \(\chi_{9802}(119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1087}{1092}\right)\) | \(e\left(\frac{191}{364}\right)\) | \(e\left(\frac{737}{1092}\right)\) | \(e\left(\frac{541}{546}\right)\) | \(e\left(\frac{139}{273}\right)\) | \(e\left(\frac{142}{273}\right)\) | \(e\left(\frac{83}{156}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{61}{91}\right)\) | \(e\left(\frac{17}{42}\right)\) |
| \(\chi_{9802}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{353}{1092}\right)\) | \(e\left(\frac{129}{364}\right)\) | \(e\left(\frac{1039}{1092}\right)\) | \(e\left(\frac{353}{546}\right)\) | \(e\left(\frac{233}{273}\right)\) | \(e\left(\frac{185}{273}\right)\) | \(e\left(\frac{37}{156}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{25}{91}\right)\) | \(e\left(\frac{1}{42}\right)\) |
| \(\chi_{9802}(189,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{617}{1092}\right)\) | \(e\left(\frac{309}{364}\right)\) | \(e\left(\frac{127}{1092}\right)\) | \(e\left(\frac{71}{546}\right)\) | \(e\left(\frac{101}{273}\right)\) | \(e\left(\frac{113}{273}\right)\) | \(e\left(\frac{85}{156}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{62}{91}\right)\) | \(e\left(\frac{19}{42}\right)\) |
| \(\chi_{9802}(253,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{263}{1092}\right)\) | \(e\left(\frac{291}{364}\right)\) | \(e\left(\frac{109}{1092}\right)\) | \(e\left(\frac{263}{546}\right)\) | \(e\left(\frac{5}{273}\right)\) | \(e\left(\frac{11}{273}\right)\) | \(e\left(\frac{127}{156}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{31}{91}\right)\) | \(e\left(\frac{13}{42}\right)\) |
| \(\chi_{9802}(271,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{565}{1092}\right)\) | \(e\left(\frac{257}{364}\right)\) | \(e\left(\frac{803}{1092}\right)\) | \(e\left(\frac{19}{546}\right)\) | \(e\left(\frac{127}{273}\right)\) | \(e\left(\frac{61}{273}\right)\) | \(e\left(\frac{137}{156}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{23}{91}\right)\) | \(e\left(\frac{11}{42}\right)\) |
| \(\chi_{9802}(275,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{547}{1092}\right)\) | \(e\left(\frac{71}{364}\right)\) | \(e\left(\frac{617}{1092}\right)\) | \(e\left(\frac{1}{546}\right)\) | \(e\left(\frac{136}{273}\right)\) | \(e\left(\frac{190}{273}\right)\) | \(e\left(\frac{155}{156}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{6}{91}\right)\) | \(e\left(\frac{5}{42}\right)\) |
| \(\chi_{9802}(301,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{535}{1092}\right)\) | \(e\left(\frac{311}{364}\right)\) | \(e\left(\frac{857}{1092}\right)\) | \(e\left(\frac{535}{546}\right)\) | \(e\left(\frac{142}{273}\right)\) | \(e\left(\frac{94}{273}\right)\) | \(e\left(\frac{11}{156}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{25}{91}\right)\) | \(e\left(\frac{29}{42}\right)\) |
| \(\chi_{9802}(345,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{485}{1092}\right)\) | \(e\left(\frac{37}{364}\right)\) | \(e\left(\frac{583}{1092}\right)\) | \(e\left(\frac{485}{546}\right)\) | \(e\left(\frac{167}{273}\right)\) | \(e\left(\frac{149}{273}\right)\) | \(e\left(\frac{61}{156}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{89}{91}\right)\) | \(e\left(\frac{31}{42}\right)\) |
| \(\chi_{9802}(375,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{769}{1092}\right)\) | \(e\left(\frac{181}{364}\right)\) | \(e\left(\frac{1091}{1092}\right)\) | \(e\left(\frac{223}{546}\right)\) | \(e\left(\frac{25}{273}\right)\) | \(e\left(\frac{55}{273}\right)\) | \(e\left(\frac{89}{156}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{64}{91}\right)\) | \(e\left(\frac{23}{42}\right)\) |
| \(\chi_{9802}(379,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{811}{1092}\right)\) | \(e\left(\frac{251}{364}\right)\) | \(e\left(\frac{797}{1092}\right)\) | \(e\left(\frac{265}{546}\right)\) | \(e\left(\frac{4}{273}\right)\) | \(e\left(\frac{118}{273}\right)\) | \(e\left(\frac{47}{156}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{43}{91}\right)\) | \(e\left(\frac{23}{42}\right)\) |
| \(\chi_{9802}(409,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{863}{1092}\right)\) | \(e\left(\frac{303}{364}\right)\) | \(e\left(\frac{121}{1092}\right)\) | \(e\left(\frac{317}{546}\right)\) | \(e\left(\frac{251}{273}\right)\) | \(e\left(\frac{170}{273}\right)\) | \(e\left(\frac{151}{156}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{82}{91}\right)\) | \(e\left(\frac{31}{42}\right)\) |
| \(\chi_{9802}(453,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{493}{1092}\right)\) | \(e\left(\frac{241}{364}\right)\) | \(e\left(\frac{59}{1092}\right)\) | \(e\left(\frac{493}{546}\right)\) | \(e\left(\frac{163}{273}\right)\) | \(e\left(\frac{31}{273}\right)\) | \(e\left(\frac{53}{156}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{46}{91}\right)\) | \(e\left(\frac{29}{42}\right)\) |
| \(\chi_{9802}(479,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{505}{1092}\right)\) | \(e\left(\frac{1}{364}\right)\) | \(e\left(\frac{911}{1092}\right)\) | \(e\left(\frac{505}{546}\right)\) | \(e\left(\frac{157}{273}\right)\) | \(e\left(\frac{127}{273}\right)\) | \(e\left(\frac{41}{156}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{27}{91}\right)\) | \(e\left(\frac{5}{42}\right)\) |
| \(\chi_{9802}(483,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{607}{1092}\right)\) | \(e\left(\frac{327}{364}\right)\) | \(e\left(\frac{509}{1092}\right)\) | \(e\left(\frac{61}{546}\right)\) | \(e\left(\frac{106}{273}\right)\) | \(e\left(\frac{124}{273}\right)\) | \(e\left(\frac{95}{156}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{2}{91}\right)\) | \(e\left(\frac{11}{42}\right)\) |
| \(\chi_{9802}(501,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{977}{1092}\right)\) | \(e\left(\frac{25}{364}\right)\) | \(e\left(\frac{571}{1092}\right)\) | \(e\left(\frac{431}{546}\right)\) | \(e\left(\frac{194}{273}\right)\) | \(e\left(\frac{263}{273}\right)\) | \(e\left(\frac{37}{156}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{38}{91}\right)\) | \(e\left(\frac{13}{42}\right)\) |
| \(\chi_{9802}(565,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{995}{1092}\right)\) | \(e\left(\frac{211}{364}\right)\) | \(e\left(\frac{757}{1092}\right)\) | \(e\left(\frac{449}{546}\right)\) | \(e\left(\frac{185}{273}\right)\) | \(e\left(\frac{134}{273}\right)\) | \(e\left(\frac{19}{156}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{55}{91}\right)\) | \(e\left(\frac{19}{42}\right)\) |
| \(\chi_{9802}(591,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{731}{1092}\right)\) | \(e\left(\frac{31}{364}\right)\) | \(e\left(\frac{577}{1092}\right)\) | \(e\left(\frac{185}{546}\right)\) | \(e\left(\frac{44}{273}\right)\) | \(e\left(\frac{206}{273}\right)\) | \(e\left(\frac{127}{156}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{18}{91}\right)\) | \(e\left(\frac{1}{42}\right)\) |
| \(\chi_{9802}(635,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1045}{1092}\right)\) | \(e\left(\frac{121}{364}\right)\) | \(e\left(\frac{1031}{1092}\right)\) | \(e\left(\frac{499}{546}\right)\) | \(e\left(\frac{160}{273}\right)\) | \(e\left(\frac{79}{273}\right)\) | \(e\left(\frac{125}{156}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{82}{91}\right)\) | \(e\left(\frac{17}{42}\right)\) |
| \(\chi_{9802}(669,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{251}{1092}\right)\) | \(e\left(\frac{167}{364}\right)\) | \(e\left(\frac{349}{1092}\right)\) | \(e\left(\frac{251}{546}\right)\) | \(e\left(\frac{11}{273}\right)\) | \(e\left(\frac{188}{273}\right)\) | \(e\left(\frac{139}{156}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{50}{91}\right)\) | \(e\left(\frac{37}{42}\right)\) |
| \(\chi_{9802}(717,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{655}{1092}\right)\) | \(e\left(\frac{95}{364}\right)\) | \(e\left(\frac{641}{1092}\right)\) | \(e\left(\frac{109}{546}\right)\) | \(e\left(\frac{82}{273}\right)\) | \(e\left(\frac{235}{273}\right)\) | \(e\left(\frac{47}{156}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{17}{91}\right)\) | \(e\left(\frac{41}{42}\right)\) |
| \(\chi_{9802}(735,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1013}{1092}\right)\) | \(e\left(\frac{33}{364}\right)\) | \(e\left(\frac{943}{1092}\right)\) | \(e\left(\frac{467}{546}\right)\) | \(e\left(\frac{176}{273}\right)\) | \(e\left(\frac{5}{273}\right)\) | \(e\left(\frac{1}{156}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{72}{91}\right)\) | \(e\left(\frac{25}{42}\right)\) |
| \(\chi_{9802}(773,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{1092}\right)\) | \(e\left(\frac{19}{364}\right)\) | \(e\left(\frac{565}{1092}\right)\) | \(e\left(\frac{131}{546}\right)\) | \(e\left(\frac{71}{273}\right)\) | \(e\left(\frac{47}{273}\right)\) | \(e\left(\frac{103}{156}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{58}{91}\right)\) | \(e\left(\frac{25}{42}\right)\) |
| \(\chi_{9802}(791,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{1092}\right)\) | \(e\left(\frac{137}{364}\right)\) | \(e\left(\frac{683}{1092}\right)\) | \(e\left(\frac{25}{546}\right)\) | \(e\left(\frac{124}{273}\right)\) | \(e\left(\frac{109}{273}\right)\) | \(e\left(\frac{53}{156}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{59}{91}\right)\) | \(e\left(\frac{41}{42}\right)\) |
| \(\chi_{9802}(839,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{1092}\right)\) | \(e\left(\frac{181}{364}\right)\) | \(e\left(\frac{727}{1092}\right)\) | \(e\left(\frac{41}{546}\right)\) | \(e\left(\frac{116}{273}\right)\) | \(e\left(\frac{146}{273}\right)\) | \(e\left(\frac{37}{156}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{64}{91}\right)\) | \(e\left(\frac{37}{42}\right)\) |
| \(\chi_{9802}(873,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{583}{1092}\right)\) | \(e\left(\frac{79}{364}\right)\) | \(e\left(\frac{989}{1092}\right)\) | \(e\left(\frac{37}{546}\right)\) | \(e\left(\frac{118}{273}\right)\) | \(e\left(\frac{205}{273}\right)\) | \(e\left(\frac{119}{156}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{40}{91}\right)\) | \(e\left(\frac{17}{42}\right)\) |
| \(\chi_{9802}(917,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{521}{1092}\right)\) | \(e\left(\frac{45}{364}\right)\) | \(e\left(\frac{955}{1092}\right)\) | \(e\left(\frac{521}{546}\right)\) | \(e\left(\frac{149}{273}\right)\) | \(e\left(\frac{164}{273}\right)\) | \(e\left(\frac{25}{156}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{32}{91}\right)\) | \(e\left(\frac{1}{42}\right)\) |
| \(\chi_{9802}(943,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{785}{1092}\right)\) | \(e\left(\frac{225}{364}\right)\) | \(e\left(\frac{43}{1092}\right)\) | \(e\left(\frac{239}{546}\right)\) | \(e\left(\frac{17}{273}\right)\) | \(e\left(\frac{92}{273}\right)\) | \(e\left(\frac{73}{156}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{69}{91}\right)\) | \(e\left(\frac{19}{42}\right)\) |
| \(\chi_{9802}(1007,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{1092}\right)\) | \(e\left(\frac{11}{364}\right)\) | \(e\left(\frac{193}{1092}\right)\) | \(e\left(\frac{95}{546}\right)\) | \(e\left(\frac{89}{273}\right)\) | \(e\left(\frac{32}{273}\right)\) | \(e\left(\frac{139}{156}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{24}{91}\right)\) | \(e\left(\frac{13}{42}\right)\) |
| \(\chi_{9802}(1025,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1069}{1092}\right)\) | \(e\left(\frac{5}{364}\right)\) | \(e\left(\frac{551}{1092}\right)\) | \(e\left(\frac{523}{546}\right)\) | \(e\left(\frac{148}{273}\right)\) | \(e\left(\frac{271}{273}\right)\) | \(e\left(\frac{101}{156}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{44}{91}\right)\) | \(e\left(\frac{11}{42}\right)\) |