from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(76230, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([110,0,55,258]))
chi.galois_orbit()
[g,chi] = znchar(Mod(31,76230))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(76230\) | |
| Conductor: | \(7623\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(330\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 7623.fh | 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: | $\Q(\zeta_{165})$ |
| Fixed field: | Number field defined by a degree 330 polynomial (not computed) |
First 31 of 80 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{76230}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{330}\right)\) | \(e\left(\frac{157}{330}\right)\) | \(e\left(\frac{239}{330}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{103}{165}\right)\) | \(e\left(\frac{23}{330}\right)\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{49}{330}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{259}{330}\right)\) |
| \(\chi_{76230}(691,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{233}{330}\right)\) | \(e\left(\frac{137}{330}\right)\) | \(e\left(\frac{259}{330}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{163}{330}\right)\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{89}{330}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{329}{330}\right)\) |
| \(\chi_{76230}(1951,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{330}\right)\) | \(e\left(\frac{329}{330}\right)\) | \(e\left(\frac{133}{330}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{116}{165}\right)\) | \(e\left(\frac{271}{330}\right)\) | \(e\left(\frac{86}{165}\right)\) | \(e\left(\frac{233}{330}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{53}{330}\right)\) |
| \(\chi_{76230}(3811,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{247}{330}\right)\) | \(e\left(\frac{73}{330}\right)\) | \(e\left(\frac{191}{330}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{17}{330}\right)\) | \(e\left(\frac{157}{165}\right)\) | \(e\left(\frac{151}{330}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{91}{330}\right)\) |
| \(\chi_{76230}(4471,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{330}\right)\) | \(e\left(\frac{323}{330}\right)\) | \(e\left(\frac{271}{330}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{152}{165}\right)\) | \(e\left(\frac{247}{330}\right)\) | \(e\left(\frac{107}{165}\right)\) | \(e\left(\frac{311}{330}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{41}{330}\right)\) |
| \(\chi_{76230}(5701,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{169}{330}\right)\) | \(e\left(\frac{241}{330}\right)\) | \(e\left(\frac{287}{330}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{29}{330}\right)\) | \(e\left(\frac{64}{165}\right)\) | \(e\left(\frac{277}{330}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{97}{330}\right)\) |
| \(\chi_{76230}(6361,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{299}{330}\right)\) | \(e\left(\frac{71}{330}\right)\) | \(e\left(\frac{127}{330}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{229}{330}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{287}{330}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{197}{330}\right)\) |
| \(\chi_{76230}(6961,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{223}{330}\right)\) | \(e\left(\frac{277}{330}\right)\) | \(e\left(\frac{119}{330}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{43}{165}\right)\) | \(e\left(\frac{173}{330}\right)\) | \(e\left(\frac{103}{165}\right)\) | \(e\left(\frac{139}{330}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{169}{330}\right)\) |
| \(\chi_{76230}(7621,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{330}\right)\) | \(e\left(\frac{257}{330}\right)\) | \(e\left(\frac{139}{330}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{53}{165}\right)\) | \(e\left(\frac{313}{330}\right)\) | \(e\left(\frac{8}{165}\right)\) | \(e\left(\frac{179}{330}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{239}{330}\right)\) |
| \(\chi_{76230}(8221,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{330}\right)\) | \(e\left(\frac{259}{330}\right)\) | \(e\left(\frac{203}{330}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{151}{165}\right)\) | \(e\left(\frac{101}{330}\right)\) | \(e\left(\frac{1}{165}\right)\) | \(e\left(\frac{43}{330}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{133}{330}\right)\) |
| \(\chi_{76230}(8881,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{330}\right)\) | \(e\left(\frac{269}{330}\right)\) | \(e\left(\frac{193}{330}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{146}{165}\right)\) | \(e\left(\frac{31}{330}\right)\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{23}{330}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{263}{330}\right)\) |
| \(\chi_{76230}(10741,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{307}{330}\right)\) | \(e\left(\frac{223}{330}\right)\) | \(e\left(\frac{41}{330}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{37}{165}\right)\) | \(e\left(\frac{287}{330}\right)\) | \(e\left(\frac{127}{165}\right)\) | \(e\left(\frac{181}{330}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{61}{330}\right)\) |
| \(\chi_{76230}(12631,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{330}\right)\) | \(e\left(\frac{271}{330}\right)\) | \(e\left(\frac{257}{330}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{149}{330}\right)\) | \(e\left(\frac{124}{165}\right)\) | \(e\left(\frac{217}{330}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{157}{330}\right)\) |
| \(\chi_{76230}(13291,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{330}\right)\) | \(e\left(\frac{101}{330}\right)\) | \(e\left(\frac{97}{330}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{19}{330}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{227}{330}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{257}{330}\right)\) |
| \(\chi_{76230}(13891,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{330}\right)\) | \(e\left(\frac{67}{330}\right)\) | \(e\left(\frac{329}{330}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{148}{165}\right)\) | \(e\left(\frac{323}{330}\right)\) | \(e\left(\frac{13}{165}\right)\) | \(e\left(\frac{229}{330}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{79}{330}\right)\) |
| \(\chi_{76230}(14551,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{263}{330}\right)\) | \(e\left(\frac{47}{330}\right)\) | \(e\left(\frac{19}{330}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{158}{165}\right)\) | \(e\left(\frac{133}{330}\right)\) | \(e\left(\frac{83}{165}\right)\) | \(e\left(\frac{269}{330}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{149}{330}\right)\) |
| \(\chi_{76230}(15151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{271}{330}\right)\) | \(e\left(\frac{199}{330}\right)\) | \(e\left(\frac{263}{330}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{16}{165}\right)\) | \(e\left(\frac{191}{330}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{163}{330}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{13}{330}\right)\) |
| \(\chi_{76230}(17671,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{330}\right)\) | \(e\left(\frac{43}{330}\right)\) | \(e\left(\frac{221}{330}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{127}{165}\right)\) | \(e\left(\frac{227}{330}\right)\) | \(e\left(\frac{97}{165}\right)\) | \(e\left(\frac{211}{330}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{31}{330}\right)\) |
| \(\chi_{76230}(18331,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{330}\right)\) | \(e\left(\frac{293}{330}\right)\) | \(e\left(\frac{301}{330}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{2}{165}\right)\) | \(e\left(\frac{127}{330}\right)\) | \(e\left(\frac{47}{165}\right)\) | \(e\left(\frac{41}{330}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{311}{330}\right)\) |
| \(\chi_{76230}(19561,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{259}{330}\right)\) | \(e\left(\frac{301}{330}\right)\) | \(e\left(\frac{227}{330}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{64}{165}\right)\) | \(e\left(\frac{269}{330}\right)\) | \(e\left(\frac{19}{165}\right)\) | \(e\left(\frac{157}{330}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{217}{330}\right)\) |
| \(\chi_{76230}(20221,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{330}\right)\) | \(e\left(\frac{131}{330}\right)\) | \(e\left(\frac{67}{330}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{149}{165}\right)\) | \(e\left(\frac{139}{330}\right)\) | \(e\left(\frac{119}{165}\right)\) | \(e\left(\frac{167}{330}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{317}{330}\right)\) |
| \(\chi_{76230}(21481,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{330}\right)\) | \(e\left(\frac{167}{330}\right)\) | \(e\left(\frac{229}{330}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{283}{330}\right)\) | \(e\left(\frac{158}{165}\right)\) | \(e\left(\frac{29}{330}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{59}{330}\right)\) |
| \(\chi_{76230}(22081,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{330}\right)\) | \(e\left(\frac{139}{330}\right)\) | \(e\left(\frac{323}{330}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{281}{330}\right)\) | \(e\left(\frac{91}{165}\right)\) | \(e\left(\frac{283}{330}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{223}{330}\right)\) |
| \(\chi_{76230}(22741,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{251}{330}\right)\) | \(e\left(\frac{149}{330}\right)\) | \(e\left(\frac{313}{330}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{211}{330}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{263}{330}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{23}{330}\right)\) |
| \(\chi_{76230}(24601,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{330}\right)\) | \(e\left(\frac{193}{330}\right)\) | \(e\left(\frac{71}{330}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{52}{165}\right)\) | \(e\left(\frac{167}{330}\right)\) | \(e\left(\frac{67}{165}\right)\) | \(e\left(\frac{241}{330}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{1}{330}\right)\) |
| \(\chi_{76230}(25261,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{197}{330}\right)\) | \(e\left(\frac{113}{330}\right)\) | \(e\left(\frac{151}{330}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{92}{165}\right)\) | \(e\left(\frac{67}{330}\right)\) | \(e\left(\frac{17}{165}\right)\) | \(e\left(\frac{71}{330}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{281}{330}\right)\) |
| \(\chi_{76230}(26491,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{330}\right)\) | \(e\left(\frac{1}{330}\right)\) | \(e\left(\frac{197}{330}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{49}{165}\right)\) | \(e\left(\frac{59}{330}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{97}{330}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{277}{330}\right)\) |
| \(\chi_{76230}(27151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{269}{330}\right)\) | \(e\left(\frac{161}{330}\right)\) | \(e\left(\frac{37}{330}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{134}{165}\right)\) | \(e\left(\frac{259}{330}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{107}{330}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{47}{330}\right)\) |
| \(\chi_{76230}(27751,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{330}\right)\) | \(e\left(\frac{307}{330}\right)\) | \(e\left(\frac{89}{330}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{293}{330}\right)\) | \(e\left(\frac{163}{165}\right)\) | \(e\left(\frac{79}{330}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{229}{330}\right)\) |
| \(\chi_{76230}(28411,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{293}{330}\right)\) | \(e\left(\frac{287}{330}\right)\) | \(e\left(\frac{109}{330}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{38}{165}\right)\) | \(e\left(\frac{103}{330}\right)\) | \(e\left(\frac{68}{165}\right)\) | \(e\left(\frac{119}{330}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{299}{330}\right)\) |
| \(\chi_{76230}(29011,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{330}\right)\) | \(e\left(\frac{79}{330}\right)\) | \(e\left(\frac{53}{330}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{76}{165}\right)\) | \(e\left(\frac{41}{330}\right)\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{73}{330}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{103}{330}\right)\) |