from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(132300, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([0,140,189,320]))
chi.galois_orbit()
[g,chi] = znchar(Mod(37,132300))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(132300\) | |
| Conductor: | \(11025\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(420\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 11025.iq | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{420})$ |
| Fixed field: | Number field defined by a degree 420 polynomial (not computed) |
First 31 of 96 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{132300}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{151}{420}\right)\) | \(e\left(\frac{377}{420}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{239}{420}\right)\) | \(e\left(\frac{199}{210}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{181}{420}\right)\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{55}{84}\right)\) |
| \(\chi_{132300}(613,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{401}{420}\right)\) | \(e\left(\frac{367}{420}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{109}{420}\right)\) | \(e\left(\frac{89}{210}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{311}{420}\right)\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{17}{84}\right)\) |
| \(\chi_{132300}(3637,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{105}\right)\) | \(e\left(\frac{131}{420}\right)\) | \(e\left(\frac{277}{420}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{199}{420}\right)\) | \(e\left(\frac{149}{210}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{221}{420}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{11}{84}\right)\) |
| \(\chi_{132300}(3817,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{105}\right)\) | \(e\left(\frac{127}{420}\right)\) | \(e\left(\frac{89}{420}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{23}{420}\right)\) | \(e\left(\frac{13}{210}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{397}{420}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{19}{84}\right)\) |
| \(\chi_{132300}(4573,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{349}{420}\right)\) | \(e\left(\frac{23}{420}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{341}{420}\right)\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{79}{420}\right)\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{37}{84}\right)\) |
| \(\chi_{132300}(7597,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{105}\right)\) | \(e\left(\frac{103}{420}\right)\) | \(e\left(\frac{221}{420}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{227}{420}\right)\) | \(e\left(\frac{37}{210}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{193}{420}\right)\) | \(e\left(\frac{37}{105}\right)\) | \(e\left(\frac{67}{84}\right)\) |
| \(\chi_{132300}(8173,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{209}{420}\right)\) | \(e\left(\frac{163}{420}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{61}{420}\right)\) | \(e\left(\frac{71}{210}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{359}{420}\right)\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{65}{84}\right)\) |
| \(\chi_{132300}(8353,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{73}{420}\right)\) | \(e\left(\frac{71}{420}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{377}{420}\right)\) | \(e\left(\frac{67}{210}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{43}{420}\right)\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{1}{84}\right)\) |
| \(\chi_{132300}(11197,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{105}\right)\) | \(e\left(\frac{23}{420}\right)\) | \(e\left(\frac{241}{420}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{67}{420}\right)\) | \(e\left(\frac{47}{210}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{353}{420}\right)\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{59}{84}\right)\) |
| \(\chi_{132300}(11377,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{79}{420}\right)\) | \(e\left(\frac{353}{420}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{11}{420}\right)\) | \(e\left(\frac{61}{210}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{409}{420}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{31}{84}\right)\) |
| \(\chi_{132300}(11953,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{105}\right)\) | \(e\left(\frac{113}{420}\right)\) | \(e\left(\frac{271}{420}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{37}{420}\right)\) | \(e\left(\frac{167}{210}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{383}{420}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{5}{84}\right)\) |
| \(\chi_{132300}(14977,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{179}{420}\right)\) | \(e\left(\frac{13}{420}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{211}{420}\right)\) | \(e\left(\frac{101}{210}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{209}{420}\right)\) | \(e\left(\frac{101}{105}\right)\) | \(e\left(\frac{83}{84}\right)\) |
| \(\chi_{132300}(15733,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{17}{420}\right)\) | \(e\left(\frac{379}{420}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{13}{420}\right)\) | \(e\left(\frac{53}{210}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{407}{420}\right)\) | \(e\left(\frac{53}{105}\right)\) | \(e\left(\frac{29}{84}\right)\) |
| \(\chi_{132300}(15913,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{361}{420}\right)\) | \(e\left(\frac{167}{420}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{29}{420}\right)\) | \(e\left(\frac{199}{210}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{391}{420}\right)\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{13}{84}\right)\) |
| \(\chi_{132300}(18937,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{31}{420}\right)\) | \(e\left(\frac{197}{420}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{419}{420}\right)\) | \(e\left(\frac{109}{210}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{420}\right)\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{43}{84}\right)\) |
| \(\chi_{132300}(19513,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{105}\right)\) | \(e\left(\frac{341}{420}\right)\) | \(e\left(\frac{67}{420}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{409}{420}\right)\) | \(e\left(\frac{149}{210}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{11}{420}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{53}{84}\right)\) |
| \(\chi_{132300}(22537,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{71}{420}\right)\) | \(e\left(\frac{397}{420}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{79}{420}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{341}{420}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{47}{84}\right)\) |
| \(\chi_{132300}(23473,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{229}{420}\right)\) | \(e\left(\frac{263}{420}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{101}{420}\right)\) | \(e\left(\frac{121}{210}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{319}{420}\right)\) | \(e\left(\frac{16}{105}\right)\) | \(e\left(\frac{25}{84}\right)\) |
| \(\chi_{132300}(26317,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{227}{420}\right)\) | \(e\left(\frac{169}{420}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{223}{420}\right)\) | \(e\left(\frac{53}{210}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{197}{420}\right)\) | \(e\left(\frac{53}{105}\right)\) | \(e\left(\frac{71}{84}\right)\) |
| \(\chi_{132300}(26497,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{403}{420}\right)\) | \(e\left(\frac{41}{420}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{407}{420}\right)\) | \(e\left(\frac{157}{210}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{13}{420}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{55}{84}\right)\) |
| \(\chi_{132300}(27073,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{74}{105}\right)\) | \(e\left(\frac{149}{420}\right)\) | \(e\left(\frac{283}{420}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{361}{420}\right)\) | \(e\left(\frac{131}{210}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{59}{420}\right)\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{17}{84}\right)\) |
| \(\chi_{132300}(27253,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{373}{420}\right)\) | \(e\left(\frac{311}{420}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{137}{420}\right)\) | \(e\left(\frac{187}{210}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{283}{420}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{73}{84}\right)\) |
| \(\chi_{132300}(30097,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{383}{420}\right)\) | \(e\left(\frac{361}{420}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{367}{420}\right)\) | \(e\left(\frac{107}{210}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{53}{420}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{11}{84}\right)\) |
| \(\chi_{132300}(30277,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{105}\right)\) | \(e\left(\frac{379}{420}\right)\) | \(e\left(\frac{173}{420}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{191}{420}\right)\) | \(e\left(\frac{181}{210}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{229}{420}\right)\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{19}{84}\right)\) |
| \(\chi_{132300}(30853,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{53}{420}\right)\) | \(e\left(\frac{391}{420}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{337}{420}\right)\) | \(e\left(\frac{17}{210}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{83}{420}\right)\) | \(e\left(\frac{17}{105}\right)\) | \(e\left(\frac{41}{84}\right)\) |
| \(\chi_{132300}(31033,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{97}{420}\right)\) | \(e\left(\frac{359}{420}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{173}{420}\right)\) | \(e\left(\frac{43}{210}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{247}{420}\right)\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{37}{84}\right)\) |
| \(\chi_{132300}(34633,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{377}{420}\right)\) | \(e\left(\frac{79}{420}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{313}{420}\right)\) | \(e\left(\frac{113}{210}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{107}{420}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{65}{84}\right)\) |
| \(\chi_{132300}(34813,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{241}{420}\right)\) | \(e\left(\frac{407}{420}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{209}{420}\right)\) | \(e\left(\frac{109}{210}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{211}{420}\right)\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{1}{84}\right)\) |
| \(\chi_{132300}(37837,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{46}{105}\right)\) | \(e\left(\frac{331}{420}\right)\) | \(e\left(\frac{17}{420}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{179}{420}\right)\) | \(e\left(\frac{19}{210}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{241}{420}\right)\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{31}{84}\right)\) |
| \(\chi_{132300}(38413,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{281}{420}\right)\) | \(e\left(\frac{187}{420}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{289}{420}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{131}{420}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{5}{84}\right)\) |
| \(\chi_{132300}(41437,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{11}{420}\right)\) | \(e\left(\frac{97}{420}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{379}{420}\right)\) | \(e\left(\frac{59}{210}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{41}{420}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{83}{84}\right)\) |