from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8624, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([0,105,320,84]))
chi.galois_orbit()
[g,chi] = znchar(Mod(37,8624))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8624\) | |
| Conductor: | \(8624\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(420\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{420})$ |
| Fixed field: | Number field defined by a degree 420 polynomial (not computed) |
First 31 of 96 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8624}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{420}\right)\) | \(e\left(\frac{61}{420}\right)\) | \(e\left(\frac{47}{210}\right)\) | \(e\left(\frac{13}{140}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{61}{210}\right)\) | \(e\left(\frac{47}{140}\right)\) |
| \(\chi_{8624}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{331}{420}\right)\) | \(e\left(\frac{233}{420}\right)\) | \(e\left(\frac{121}{210}\right)\) | \(e\left(\frac{29}{140}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{37}{105}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{23}{210}\right)\) | \(e\left(\frac{51}{140}\right)\) |
| \(\chi_{8624}(93,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{269}{420}\right)\) | \(e\left(\frac{367}{420}\right)\) | \(e\left(\frac{59}{210}\right)\) | \(e\left(\frac{131}{140}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{157}{210}\right)\) | \(e\left(\frac{129}{140}\right)\) |
| \(\chi_{8624}(317,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{401}{420}\right)\) | \(e\left(\frac{163}{420}\right)\) | \(e\left(\frac{191}{210}\right)\) | \(e\left(\frac{99}{140}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{163}{210}\right)\) | \(e\left(\frac{121}{140}\right)\) |
| \(\chi_{8624}(333,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{420}\right)\) | \(e\left(\frac{179}{420}\right)\) | \(e\left(\frac{193}{210}\right)\) | \(e\left(\frac{107}{140}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{46}{105}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{179}{210}\right)\) | \(e\left(\frac{53}{140}\right)\) |
| \(\chi_{8624}(389,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{367}{420}\right)\) | \(e\left(\frac{101}{420}\right)\) | \(e\left(\frac{157}{210}\right)\) | \(e\left(\frac{33}{140}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{101}{210}\right)\) | \(e\left(\frac{87}{140}\right)\) |
| \(\chi_{8624}(445,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{289}{420}\right)\) | \(e\left(\frac{107}{420}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{71}{140}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{107}{210}\right)\) | \(e\left(\frac{9}{140}\right)\) |
| \(\chi_{8624}(597,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{420}\right)\) | \(e\left(\frac{349}{420}\right)\) | \(e\left(\frac{83}{210}\right)\) | \(e\left(\frac{17}{140}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{41}{105}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{139}{210}\right)\) | \(e\left(\frac{83}{140}\right)\) |
| \(\chi_{8624}(653,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{420}\right)\) | \(e\left(\frac{151}{420}\right)\) | \(e\left(\frac{137}{210}\right)\) | \(e\left(\frac{23}{140}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{74}{105}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{151}{210}\right)\) | \(e\left(\frac{137}{140}\right)\) |
| \(\chi_{8624}(669,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{420}\right)\) | \(e\left(\frac{383}{420}\right)\) | \(e\left(\frac{61}{210}\right)\) | \(e\left(\frac{139}{140}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{173}{210}\right)\) | \(e\left(\frac{61}{140}\right)\) |
| \(\chi_{8624}(709,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{359}{420}\right)\) | \(e\left(\frac{37}{420}\right)\) | \(e\left(\frac{149}{210}\right)\) | \(e\left(\frac{1}{140}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{23}{105}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{37}{210}\right)\) | \(e\left(\frac{79}{140}\right)\) |
| \(\chi_{8624}(933,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{420}\right)\) | \(e\left(\frac{253}{420}\right)\) | \(e\left(\frac{71}{210}\right)\) | \(e\left(\frac{109}{140}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{92}{105}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{43}{210}\right)\) | \(e\left(\frac{71}{140}\right)\) |
| \(\chi_{8624}(1005,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{420}\right)\) | \(e\left(\frac{251}{420}\right)\) | \(e\left(\frac{97}{210}\right)\) | \(e\left(\frac{3}{140}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{41}{210}\right)\) | \(e\left(\frac{97}{140}\right)\) |
| \(\chi_{8624}(1061,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{420}\right)\) | \(e\left(\frac{257}{420}\right)\) | \(e\left(\frac{19}{210}\right)\) | \(e\left(\frac{41}{140}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{103}{105}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{47}{210}\right)\) | \(e\left(\frac{19}{140}\right)\) |
| \(\chi_{8624}(1213,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{420}\right)\) | \(e\left(\frac{19}{420}\right)\) | \(e\left(\frac{173}{210}\right)\) | \(e\left(\frac{27}{140}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{19}{210}\right)\) | \(e\left(\frac{33}{140}\right)\) |
| \(\chi_{8624}(1269,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{227}{420}\right)\) | \(e\left(\frac{241}{420}\right)\) | \(e\left(\frac{17}{210}\right)\) | \(e\left(\frac{33}{140}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{87}{140}\right)\) |
| \(\chi_{8624}(1285,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{211}{420}\right)\) | \(e\left(\frac{113}{420}\right)\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{109}{140}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{22}{105}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{113}{210}\right)\) | \(e\left(\frac{71}{140}\right)\) |
| \(\chi_{8624}(1325,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{420}\right)\) | \(e\left(\frac{127}{420}\right)\) | \(e\left(\frac{29}{210}\right)\) | \(e\left(\frac{11}{140}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{127}{210}\right)\) | \(e\left(\frac{29}{140}\right)\) |
| \(\chi_{8624}(1565,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{420}\right)\) | \(e\left(\frac{59}{420}\right)\) | \(e\left(\frac{73}{210}\right)\) | \(e\left(\frac{47}{140}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{59}{210}\right)\) | \(e\left(\frac{73}{140}\right)\) |
| \(\chi_{8624}(1621,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{247}{420}\right)\) | \(e\left(\frac{401}{420}\right)\) | \(e\left(\frac{37}{210}\right)\) | \(e\left(\frac{113}{140}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{79}{105}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{191}{210}\right)\) | \(e\left(\frac{107}{140}\right)\) |
| \(\chi_{8624}(1677,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{169}{420}\right)\) | \(e\left(\frac{407}{420}\right)\) | \(e\left(\frac{169}{210}\right)\) | \(e\left(\frac{11}{140}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{197}{210}\right)\) | \(e\left(\frac{29}{140}\right)\) |
| \(\chi_{8624}(1829,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{263}{420}\right)\) | \(e\left(\frac{109}{420}\right)\) | \(e\left(\frac{53}{210}\right)\) | \(e\left(\frac{37}{140}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{109}{210}\right)\) | \(e\left(\frac{123}{140}\right)\) |
| \(\chi_{8624}(1885,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{317}{420}\right)\) | \(e\left(\frac{331}{420}\right)\) | \(e\left(\frac{107}{210}\right)\) | \(e\left(\frac{43}{140}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{121}{210}\right)\) | \(e\left(\frac{37}{140}\right)\) |
| \(\chi_{8624}(1901,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{361}{420}\right)\) | \(e\left(\frac{263}{420}\right)\) | \(e\left(\frac{151}{210}\right)\) | \(e\left(\frac{79}{140}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{53}{210}\right)\) | \(e\left(\frac{81}{140}\right)\) |
| \(\chi_{8624}(2165,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{251}{420}\right)\) | \(e\left(\frac{13}{420}\right)\) | \(e\left(\frac{41}{210}\right)\) | \(e\left(\frac{129}{140}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{13}{210}\right)\) | \(e\left(\frac{111}{140}\right)\) |
| \(\chi_{8624}(2181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{223}{420}\right)\) | \(e\left(\frac{209}{420}\right)\) | \(e\left(\frac{13}{210}\right)\) | \(e\left(\frac{17}{140}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{83}{140}\right)\) |
| \(\chi_{8624}(2237,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{397}{420}\right)\) | \(e\left(\frac{131}{420}\right)\) | \(e\left(\frac{187}{210}\right)\) | \(e\left(\frac{83}{140}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{131}{210}\right)\) | \(e\left(\frac{117}{140}\right)\) |
| \(\chi_{8624}(2293,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{319}{420}\right)\) | \(e\left(\frac{137}{420}\right)\) | \(e\left(\frac{109}{210}\right)\) | \(e\left(\frac{121}{140}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{137}{210}\right)\) | \(e\left(\frac{39}{140}\right)\) |
| \(\chi_{8624}(2445,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{353}{420}\right)\) | \(e\left(\frac{199}{420}\right)\) | \(e\left(\frac{143}{210}\right)\) | \(e\left(\frac{47}{140}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{101}{105}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{199}{210}\right)\) | \(e\left(\frac{73}{140}\right)\) |
| \(\chi_{8624}(2501,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{407}{420}\right)\) | \(e\left(\frac{1}{420}\right)\) | \(e\left(\frac{197}{210}\right)\) | \(e\left(\frac{53}{140}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{127}{140}\right)\) |
| \(\chi_{8624}(2557,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{209}{420}\right)\) | \(e\left(\frac{307}{420}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{31}{140}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{97}{210}\right)\) | \(e\left(\frac{69}{140}\right)\) |