from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(76230, base_ring=CyclotomicField(660))
M = H._module
chi = DirichletCharacter(H, M([550,495,0,348]))
chi.galois_orbit()
[g,chi] = znchar(Mod(113,76230))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(76230\) | |
| Conductor: | \(5445\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(660\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 5445.do | 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: | $\Q(\zeta_{660})$ |
| Fixed field: | Number field defined by a degree 660 polynomial (not computed) |
First 31 of 160 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{76230}(113,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{660}\right)\) | \(e\left(\frac{19}{220}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{43}{132}\right)\) | \(e\left(\frac{49}{165}\right)\) | \(e\left(\frac{2}{165}\right)\) | \(e\left(\frac{197}{220}\right)\) | \(e\left(\frac{97}{330}\right)\) | \(e\left(\frac{101}{132}\right)\) | \(e\left(\frac{169}{660}\right)\) |
| \(\chi_{76230}(533,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{289}{660}\right)\) | \(e\left(\frac{107}{220}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{131}{132}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{21}{220}\right)\) | \(e\left(\frac{251}{330}\right)\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{257}{660}\right)\) |
| \(\chi_{76230}(1037,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{660}\right)\) | \(e\left(\frac{29}{220}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{142}{165}\right)\) | \(e\left(\frac{127}{220}\right)\) | \(e\left(\frac{287}{330}\right)\) | \(e\left(\frac{43}{132}\right)\) | \(e\left(\frac{119}{660}\right)\) |
| \(\chi_{76230}(1247,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{647}{660}\right)\) | \(e\left(\frac{101}{220}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{85}{132}\right)\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{83}{165}\right)\) | \(e\left(\frac{63}{220}\right)\) | \(e\left(\frac{313}{330}\right)\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{331}{660}\right)\) |
| \(\chi_{76230}(1373,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{660}\right)\) | \(e\left(\frac{3}{220}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{67}{132}\right)\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{119}{165}\right)\) | \(e\left(\frac{89}{220}\right)\) | \(e\left(\frac{79}{330}\right)\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{73}{660}\right)\) |
| \(\chi_{76230}(2297,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{331}{660}\right)\) | \(e\left(\frac{153}{220}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{53}{165}\right)\) | \(e\left(\frac{19}{165}\right)\) | \(e\left(\frac{139}{220}\right)\) | \(e\left(\frac{179}{330}\right)\) | \(e\left(\frac{19}{132}\right)\) | \(e\left(\frac{203}{660}\right)\) |
| \(\chi_{76230}(2633,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{660}\right)\) | \(e\left(\frac{71}{220}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{76}{165}\right)\) | \(e\left(\frac{158}{165}\right)\) | \(e\left(\frac{53}{220}\right)\) | \(e\left(\frac{73}{330}\right)\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{481}{660}\right)\) |
| \(\chi_{76230}(3557,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{367}{660}\right)\) | \(e\left(\frac{161}{220}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{146}{165}\right)\) | \(e\left(\frac{43}{165}\right)\) | \(e\left(\frac{83}{220}\right)\) | \(e\left(\frac{23}{330}\right)\) | \(e\left(\frac{43}{132}\right)\) | \(e\left(\frac{251}{660}\right)\) |
| \(\chi_{76230}(3683,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{601}{660}\right)\) | \(e\left(\frac{103}{220}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{71}{132}\right)\) | \(e\left(\frac{8}{165}\right)\) | \(e\left(\frac{34}{165}\right)\) | \(e\left(\frac{49}{220}\right)\) | \(e\left(\frac{329}{330}\right)\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{233}{660}\right)\) |
| \(\chi_{76230}(3767,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{660}\right)\) | \(e\left(\frac{117}{220}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{61}{132}\right)\) | \(e\left(\frac{157}{165}\right)\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{171}{220}\right)\) | \(e\left(\frac{1}{330}\right)\) | \(e\left(\frac{131}{132}\right)\) | \(e\left(\frac{427}{660}\right)\) |
| \(\chi_{76230}(4943,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{397}{660}\right)\) | \(e\left(\frac{131}{220}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{95}{132}\right)\) | \(e\left(\frac{86}{165}\right)\) | \(e\left(\frac{118}{165}\right)\) | \(e\left(\frac{73}{220}\right)\) | \(e\left(\frac{113}{330}\right)\) | \(e\left(\frac{85}{132}\right)\) | \(e\left(\frac{401}{660}\right)\) |
| \(\chi_{76230}(5153,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{660}\right)\) | \(e\left(\frac{27}{220}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{142}{165}\right)\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{141}{220}\right)\) | \(e\left(\frac{271}{330}\right)\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{217}{660}\right)\) |
| \(\chi_{76230}(5657,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{383}{660}\right)\) | \(e\left(\frac{189}{220}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{85}{132}\right)\) | \(e\left(\frac{4}{165}\right)\) | \(e\left(\frac{17}{165}\right)\) | \(e\left(\frac{107}{220}\right)\) | \(e\left(\frac{247}{330}\right)\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{199}{660}\right)\) |
| \(\chi_{76230}(6917,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{660}\right)\) | \(e\left(\frac{133}{220}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{37}{132}\right)\) | \(e\left(\frac{13}{165}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{59}{220}\right)\) | \(e\left(\frac{19}{330}\right)\) | \(e\left(\frac{47}{132}\right)\) | \(e\left(\frac{523}{660}\right)\) |
| \(\chi_{76230}(7043,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{533}{660}\right)\) | \(e\left(\frac{39}{220}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{79}{132}\right)\) | \(e\left(\frac{34}{165}\right)\) | \(e\left(\frac{62}{165}\right)\) | \(e\left(\frac{57}{220}\right)\) | \(e\left(\frac{37}{330}\right)\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{289}{660}\right)\) |
| \(\chi_{76230}(7463,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{409}{660}\right)\) | \(e\left(\frac{207}{220}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{47}{132}\right)\) | \(e\left(\frac{62}{165}\right)\) | \(e\left(\frac{16}{165}\right)\) | \(e\left(\frac{201}{220}\right)\) | \(e\left(\frac{281}{330}\right)\) | \(e\left(\frac{49}{132}\right)\) | \(e\left(\frac{197}{660}\right)\) |
| \(\chi_{76230}(7967,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{523}{660}\right)\) | \(e\left(\frac{49}{220}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{37}{165}\right)\) | \(e\left(\frac{207}{220}\right)\) | \(e\left(\frac{227}{330}\right)\) | \(e\left(\frac{103}{132}\right)\) | \(e\left(\frac{239}{660}\right)\) |
| \(\chi_{76230}(8177,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{467}{660}\right)\) | \(e\left(\frac{61}{220}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{1}{165}\right)\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{123}{220}\right)\) | \(e\left(\frac{103}{330}\right)\) | \(e\left(\frac{95}{132}\right)\) | \(e\left(\frac{91}{660}\right)\) |
| \(\chi_{76230}(8303,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{401}{660}\right)\) | \(e\left(\frac{83}{220}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{79}{132}\right)\) | \(e\left(\frac{133}{165}\right)\) | \(e\left(\frac{29}{165}\right)\) | \(e\left(\frac{189}{220}\right)\) | \(e\left(\frac{169}{330}\right)\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{553}{660}\right)\) |
| \(\chi_{76230}(9227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{660}\right)\) | \(e\left(\frac{13}{220}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{41}{132}\right)\) | \(e\left(\frac{158}{165}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{19}{220}\right)\) | \(e\left(\frac{269}{330}\right)\) | \(e\left(\frac{127}{132}\right)\) | \(e\left(\frac{23}{660}\right)\) |
| \(\chi_{76230}(9353,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{660}\right)\) | \(e\left(\frac{119}{220}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{47}{132}\right)\) | \(e\left(\frac{29}{165}\right)\) | \(e\left(\frac{82}{165}\right)\) | \(e\left(\frac{157}{220}\right)\) | \(e\left(\frac{17}{330}\right)\) | \(e\left(\frac{49}{132}\right)\) | \(e\left(\frac{329}{660}\right)\) |
| \(\chi_{76230}(9563,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{497}{660}\right)\) | \(e\left(\frac{31}{220}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{91}{132}\right)\) | \(e\left(\frac{106}{165}\right)\) | \(e\left(\frac{38}{165}\right)\) | \(e\left(\frac{113}{220}\right)\) | \(e\left(\frac{193}{330}\right)\) | \(e\left(\frac{5}{132}\right)\) | \(e\left(\frac{241}{660}\right)\) |
| \(\chi_{76230}(10613,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{301}{660}\right)\) | \(e\left(\frac{183}{220}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{149}{220}\right)\) | \(e\left(\frac{89}{330}\right)\) | \(e\left(\frac{109}{132}\right)\) | \(e\left(\frac{53}{660}\right)\) |
| \(\chi_{76230}(10697,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{660}\right)\) | \(e\left(\frac{217}{220}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{109}{132}\right)\) | \(e\left(\frac{82}{165}\right)\) | \(e\left(\frac{101}{165}\right)\) | \(e\left(\frac{131}{220}\right)\) | \(e\left(\frac{31}{330}\right)\) | \(e\left(\frac{35}{132}\right)\) | \(e\left(\frac{367}{660}\right)\) |
| \(\chi_{76230}(11873,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{217}{660}\right)\) | \(e\left(\frac{91}{220}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{23}{132}\right)\) | \(e\left(\frac{116}{165}\right)\) | \(e\left(\frac{163}{165}\right)\) | \(e\left(\frac{133}{220}\right)\) | \(e\left(\frac{233}{330}\right)\) | \(e\left(\frac{97}{132}\right)\) | \(e\left(\frac{161}{660}\right)\) |
| \(\chi_{76230}(12083,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{269}{660}\right)\) | \(e\left(\frac{127}{220}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{79}{132}\right)\) | \(e\left(\frac{67}{165}\right)\) | \(e\left(\frac{161}{165}\right)\) | \(e\left(\frac{101}{220}\right)\) | \(e\left(\frac{301}{330}\right)\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{157}{660}\right)\) |
| \(\chi_{76230}(13007,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{439}{660}\right)\) | \(e\left(\frac{177}{220}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{2}{165}\right)\) | \(e\left(\frac{91}{165}\right)\) | \(e\left(\frac{191}{220}\right)\) | \(e\left(\frac{41}{330}\right)\) | \(e\left(\frac{91}{132}\right)\) | \(e\left(\frac{347}{660}\right)\) |
| \(\chi_{76230}(13847,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{491}{660}\right)\) | \(e\left(\frac{213}{220}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{49}{132}\right)\) | \(e\left(\frac{118}{165}\right)\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{159}{220}\right)\) | \(e\left(\frac{109}{330}\right)\) | \(e\left(\frac{23}{132}\right)\) | \(e\left(\frac{343}{660}\right)\) |
| \(\chi_{76230}(13973,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{293}{660}\right)\) | \(e\left(\frac{59}{220}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{115}{132}\right)\) | \(e\left(\frac{19}{165}\right)\) | \(e\left(\frac{122}{165}\right)\) | \(e\left(\frac{137}{220}\right)\) | \(e\left(\frac{307}{330}\right)\) | \(e\left(\frac{89}{132}\right)\) | \(e\left(\frac{409}{660}\right)\) |
| \(\chi_{76230}(14393,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{529}{660}\right)\) | \(e\left(\frac{87}{220}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{95}{132}\right)\) | \(e\left(\frac{152}{165}\right)\) | \(e\left(\frac{151}{165}\right)\) | \(e\left(\frac{161}{220}\right)\) | \(e\left(\frac{311}{330}\right)\) | \(e\left(\frac{85}{132}\right)\) | \(e\left(\frac{137}{660}\right)\) |
| \(\chi_{76230}(14897,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{283}{660}\right)\) | \(e\left(\frac{69}{220}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{89}{132}\right)\) | \(e\left(\frac{149}{165}\right)\) | \(e\left(\frac{97}{165}\right)\) | \(e\left(\frac{67}{220}\right)\) | \(e\left(\frac{167}{330}\right)\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{359}{660}\right)\) |