from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(132300, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([0,280,231,370]))
chi.galois_orbit()
[g,chi] = znchar(Mod(73,132300))
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.ir | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | 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 | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{132300}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{74}{105}\right)\) | \(e\left(\frac{359}{420}\right)\) | \(e\left(\frac{73}{420}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{361}{420}\right)\) | \(e\left(\frac{131}{210}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{59}{420}\right)\) | \(e\left(\frac{157}{210}\right)\) | \(e\left(\frac{17}{84}\right)\) |
| \(\chi_{132300}(2413,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{151}{420}\right)\) | \(e\left(\frac{377}{420}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{29}{420}\right)\) | \(e\left(\frac{199}{210}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{391}{420}\right)\) | \(e\left(\frac{83}{210}\right)\) | \(e\left(\frac{13}{84}\right)\) |
| \(\chi_{132300}(3097,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{293}{420}\right)\) | \(e\left(\frac{331}{420}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{187}{420}\right)\) | \(e\left(\frac{197}{210}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{233}{420}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{23}{84}\right)\) |
| \(\chi_{132300}(5437,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{61}{420}\right)\) | \(e\left(\frac{347}{420}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{59}{420}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{361}{420}\right)\) | \(e\left(\frac{53}{210}\right)\) | \(e\left(\frac{67}{84}\right)\) |
| \(\chi_{132300}(6877,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{389}{420}\right)\) | \(e\left(\frac{223}{420}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{211}{420}\right)\) | \(e\left(\frac{101}{210}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{209}{420}\right)\) | \(e\left(\frac{97}{210}\right)\) | \(e\left(\frac{83}{84}\right)\) |
| \(\chi_{132300}(7633,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{105}\right)\) | \(e\left(\frac{47}{420}\right)\) | \(e\left(\frac{109}{420}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{73}{420}\right)\) | \(e\left(\frac{23}{210}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{347}{420}\right)\) | \(e\left(\frac{151}{210}\right)\) | \(e\left(\frac{53}{84}\right)\) |
| \(\chi_{132300}(9217,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{105}\right)\) | \(e\left(\frac{337}{420}\right)\) | \(e\left(\frac{299}{420}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{23}{420}\right)\) | \(e\left(\frac{13}{210}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{397}{420}\right)\) | \(e\left(\frac{131}{210}\right)\) | \(e\left(\frac{19}{84}\right)\) |
| \(\chi_{132300}(9973,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{199}{420}\right)\) | \(e\left(\frac{113}{420}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{41}{420}\right)\) | \(e\left(\frac{151}{210}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{379}{420}\right)\) | \(e\left(\frac{197}{210}\right)\) | \(e\left(\frac{1}{84}\right)\) |
| \(\chi_{132300}(11413,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{105}\right)\) | \(e\left(\frac{311}{420}\right)\) | \(e\left(\frac{337}{420}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{349}{420}\right)\) | \(e\left(\frac{179}{210}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{71}{420}\right)\) | \(e\left(\frac{43}{210}\right)\) | \(e\left(\frac{29}{84}\right)\) |
| \(\chi_{132300}(12997,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{193}{420}\right)\) | \(e\left(\frac{251}{420}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{407}{420}\right)\) | \(e\left(\frac{157}{210}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{13}{420}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{55}{84}\right)\) |
| \(\chi_{132300}(13753,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{105}\right)\) | \(e\left(\frac{223}{420}\right)\) | \(e\left(\frac{401}{420}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{257}{420}\right)\) | \(e\left(\frac{127}{210}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{163}{420}\right)\) | \(e\left(\frac{149}{210}\right)\) | \(e\left(\frac{37}{84}\right)\) |
| \(\chi_{132300}(17533,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{247}{420}\right)\) | \(e\left(\frac{269}{420}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{53}{420}\right)\) | \(e\left(\frac{103}{210}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{367}{420}\right)\) | \(e\left(\frac{101}{210}\right)\) | \(e\left(\frac{73}{84}\right)\) |
| \(\chi_{132300}(18217,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{105}\right)\) | \(e\left(\frac{257}{420}\right)\) | \(e\left(\frac{319}{420}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{283}{420}\right)\) | \(e\left(\frac{23}{210}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{137}{420}\right)\) | \(e\left(\frac{151}{210}\right)\) | \(e\left(\frac{11}{84}\right)\) |
| \(\chi_{132300}(18973,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{419}{420}\right)\) | \(e\left(\frac{373}{420}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{61}{420}\right)\) | \(e\left(\frac{71}{210}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{359}{420}\right)\) | \(e\left(\frac{37}{210}\right)\) | \(e\left(\frac{65}{84}\right)\) |
| \(\chi_{132300}(21313,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{271}{420}\right)\) | \(e\left(\frac{137}{420}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{269}{420}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{151}{420}\right)\) | \(e\left(\frac{53}{210}\right)\) | \(e\left(\frac{25}{84}\right)\) |
| \(\chi_{132300}(21997,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{353}{420}\right)\) | \(e\left(\frac{211}{420}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{307}{420}\right)\) | \(e\left(\frac{137}{210}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{113}{420}\right)\) | \(e\left(\frac{169}{210}\right)\) | \(e\left(\frac{71}{84}\right)\) |
| \(\chi_{132300}(22753,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{263}{420}\right)\) | \(e\left(\frac{181}{420}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{337}{420}\right)\) | \(e\left(\frac{17}{210}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{83}{420}\right)\) | \(e\left(\frac{139}{210}\right)\) | \(e\left(\frac{41}{84}\right)\) |
| \(\chi_{132300}(24337,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{181}{420}\right)\) | \(e\left(\frac{107}{420}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{299}{420}\right)\) | \(e\left(\frac{169}{210}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{121}{420}\right)\) | \(e\left(\frac{23}{210}\right)\) | \(e\left(\frac{79}{84}\right)\) |
| \(\chi_{132300}(25777,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{29}{420}\right)\) | \(e\left(\frac{103}{420}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{331}{420}\right)\) | \(e\left(\frac{41}{210}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{89}{420}\right)\) | \(e\left(\frac{187}{210}\right)\) | \(e\left(\frac{47}{84}\right)\) |
| \(\chi_{132300}(26533,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{107}{420}\right)\) | \(e\left(\frac{409}{420}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{193}{420}\right)\) | \(e\left(\frac{173}{210}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{227}{420}\right)\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{17}{84}\right)\) |
| \(\chi_{132300}(28117,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{37}{420}\right)\) | \(e\left(\frac{59}{420}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{263}{420}\right)\) | \(e\left(\frac{103}{210}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{157}{420}\right)\) | \(e\left(\frac{101}{210}\right)\) | \(e\left(\frac{31}{84}\right)\) |
| \(\chi_{132300}(28873,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{64}{105}\right)\) | \(e\left(\frac{319}{420}\right)\) | \(e\left(\frac{293}{420}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{281}{420}\right)\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{139}{420}\right)\) | \(e\left(\frac{167}{210}\right)\) | \(e\left(\frac{13}{84}\right)\) |
| \(\chi_{132300}(31897,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{105}\right)\) | \(e\left(\frac{313}{420}\right)\) | \(e\left(\frac{11}{420}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{227}{420}\right)\) | \(e\left(\frac{37}{210}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{193}{420}\right)\) | \(e\left(\frac{179}{210}\right)\) | \(e\left(\frac{67}{84}\right)\) |
| \(\chi_{132300}(33337,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{221}{420}\right)\) | \(e\left(\frac{307}{420}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{379}{420}\right)\) | \(e\left(\frac{59}{210}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{41}{420}\right)\) | \(e\left(\frac{13}{210}\right)\) | \(e\left(\frac{83}{84}\right)\) |
| \(\chi_{132300}(35677,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{105}\right)\) | \(e\left(\frac{169}{420}\right)\) | \(e\left(\frac{383}{420}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{191}{420}\right)\) | \(e\left(\frac{181}{210}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{229}{420}\right)\) | \(e\left(\frac{47}{210}\right)\) | \(e\left(\frac{19}{84}\right)\) |
| \(\chi_{132300}(36433,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{367}{420}\right)\) | \(e\left(\frac{29}{420}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{293}{420}\right)\) | \(e\left(\frac{193}{210}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{127}{420}\right)\) | \(e\left(\frac{71}{210}\right)\) | \(e\left(\frac{1}{84}\right)\) |
| \(\chi_{132300}(37117,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{317}{420}\right)\) | \(e\left(\frac{199}{420}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{403}{420}\right)\) | \(e\left(\frac{173}{210}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{17}{420}\right)\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{59}{84}\right)\) |
| \(\chi_{132300}(37873,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{59}{420}\right)\) | \(e\left(\frac{253}{420}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{181}{420}\right)\) | \(e\left(\frac{11}{210}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{239}{420}\right)\) | \(e\left(\frac{127}{210}\right)\) | \(e\left(\frac{29}{84}\right)\) |
| \(\chi_{132300}(40213,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{391}{420}\right)\) | \(e\left(\frac{317}{420}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{89}{420}\right)\) | \(e\left(\frac{169}{210}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{331}{420}\right)\) | \(e\left(\frac{23}{210}\right)\) | \(e\left(\frac{37}{84}\right)\) |
| \(\chi_{132300}(41653,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{105}\right)\) | \(e\left(\frac{323}{420}\right)\) | \(e\left(\frac{61}{420}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{37}{420}\right)\) | \(e\left(\frac{167}{210}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{383}{420}\right)\) | \(e\left(\frac{19}{210}\right)\) | \(e\left(\frac{5}{84}\right)\) |
| \(\chi_{132300}(44677,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{89}{420}\right)\) | \(e\left(\frac{403}{420}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{31}{420}\right)\) | \(e\left(\frac{191}{210}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{389}{420}\right)\) | \(e\left(\frac{67}{210}\right)\) | \(e\left(\frac{11}{84}\right)\) |