from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1960, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([0,42,21,64]))
chi.galois_orbit()
[g,chi] = znchar(Mod(37,1960))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(1960\) | |
Conductor: | \(1960\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(84\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | 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 | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{1960}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{1960}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{1960}(93,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{1960}(277,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{1960}(317,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{1960}(333,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{1960}(597,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{1960}(613,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{1960}(653,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{1960}(837,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{1960}(877,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{1960}(893,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{1960}(933,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{1960}(1117,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{1960}(1173,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{1960}(1213,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{1960}(1397,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{1960}(1437,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{1960}(1453,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{1960}(1493,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{1960}(1677,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{1960}(1717,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{1960}(1773,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{1960}(1957,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{2}{3}\right)\) |