sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(2205, base_ring=CyclotomicField(84))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,21,64]))
sage: chi.galois_orbit()
pari: [g,chi] = znchar(Mod(37,2205))
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2205\) | |
| Conductor: | \(245\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(84\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 245.w | 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: | $\Q(\zeta_{84})$ |
| Fixed field: | Number field defined by a degree 84 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2205}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{59}{84}\right)\) |
| \(\chi_{2205}(163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{29}{84}\right)\) |
| \(\chi_{2205}(172,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{19}{84}\right)\) |
| \(\chi_{2205}(298,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{25}{84}\right)\) |
| \(\chi_{2205}(352,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{47}{84}\right)\) |
| \(\chi_{2205}(478,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{17}{84}\right)\) |
| \(\chi_{2205}(487,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{55}{84}\right)\) |
| \(\chi_{2205}(613,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{61}{84}\right)\) |
| \(\chi_{2205}(793,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{5}{84}\right)\) |
| \(\chi_{2205}(928,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{13}{84}\right)\) |
| \(\chi_{2205}(982,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{23}{84}\right)\) |
| \(\chi_{2205}(1117,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{43}{84}\right)\) |
| \(\chi_{2205}(1297,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{11}{84}\right)\) |
| \(\chi_{2205}(1423,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{65}{84}\right)\) |
| \(\chi_{2205}(1432,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{79}{84}\right)\) |
| \(\chi_{2205}(1558,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{1}{84}\right)\) |
| \(\chi_{2205}(1612,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{83}{84}\right)\) |
| \(\chi_{2205}(1738,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{53}{84}\right)\) |
| \(\chi_{2205}(1747,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{31}{84}\right)\) |
| \(\chi_{2205}(1873,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{37}{84}\right)\) |
| \(\chi_{2205}(1927,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{71}{84}\right)\) |
| \(\chi_{2205}(2053,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{41}{84}\right)\) |
| \(\chi_{2205}(2062,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{67}{84}\right)\) |
| \(\chi_{2205}(2188,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{73}{84}\right)\) |