from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3204, base_ring=CyclotomicField(88))
M = H._module
chi = DirichletCharacter(H, M([0,0,41]))
chi.galois_orbit()
[g,chi] = znchar(Mod(145,3204))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(3204\) | |
Conductor: | \(89\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(88\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 89.h | 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: | $\Q(\zeta_{88})$ |
Fixed field: | Number field defined by a degree 88 polynomial |
First 31 of 40 characters in Galois orbit
Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{3204}(145,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{65}{88}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{63}{88}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{27}{88}\right)\) | \(e\left(\frac{49}{88}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{43}{88}\right)\) | \(e\left(\frac{39}{88}\right)\) |
\(\chi_{3204}(181,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{81}{88}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{23}{88}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{35}{88}\right)\) | \(e\left(\frac{57}{88}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{59}{88}\right)\) | \(e\left(\frac{31}{88}\right)\) |
\(\chi_{3204}(253,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{69}{88}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{75}{88}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{7}{88}\right)\) | \(e\left(\frac{29}{88}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{47}{88}\right)\) | \(e\left(\frac{59}{88}\right)\) |
\(\chi_{3204}(325,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{3}{88}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{53}{88}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{73}{88}\right)\) | \(e\left(\frac{51}{88}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{25}{88}\right)\) | \(e\left(\frac{37}{88}\right)\) |
\(\chi_{3204}(397,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{29}{88}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{43}{88}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{31}{88}\right)\) | \(e\left(\frac{53}{88}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{7}{88}\right)\) | \(e\left(\frac{35}{88}\right)\) |
\(\chi_{3204}(469,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{9}{88}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{71}{88}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{43}{88}\right)\) | \(e\left(\frac{65}{88}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{75}{88}\right)\) | \(e\left(\frac{23}{88}\right)\) |
\(\chi_{3204}(505,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{71}{88}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{81}{88}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{85}{88}\right)\) | \(e\left(\frac{63}{88}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{5}{88}\right)\) | \(e\left(\frac{25}{88}\right)\) |
\(\chi_{3204}(541,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{49}{88}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{15}{88}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{19}{88}\right)\) | \(e\left(\frac{41}{88}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{27}{88}\right)\) | \(e\left(\frac{47}{88}\right)\) |
\(\chi_{3204}(577,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{61}{88}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{51}{88}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{47}{88}\right)\) | \(e\left(\frac{69}{88}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{39}{88}\right)\) | \(e\left(\frac{19}{88}\right)\) |
\(\chi_{3204}(649,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{79}{88}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{17}{88}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{45}{88}\right)\) | \(e\left(\frac{23}{88}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{13}{88}\right)\) | \(e\left(\frac{65}{88}\right)\) |
\(\chi_{3204}(685,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{23}{88}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{25}{88}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{61}{88}\right)\) | \(e\left(\frac{39}{88}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{45}{88}\right)\) | \(e\left(\frac{49}{88}\right)\) |
\(\chi_{3204}(829,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{1}{88}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{47}{88}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{83}{88}\right)\) | \(e\left(\frac{17}{88}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{67}{88}\right)\) | \(e\left(\frac{71}{88}\right)\) |
\(\chi_{3204}(973,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{13}{88}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{83}{88}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{23}{88}\right)\) | \(e\left(\frac{45}{88}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{79}{88}\right)\) | \(e\left(\frac{43}{88}\right)\) |
\(\chi_{3204}(1009,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{7}{88}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{65}{88}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{53}{88}\right)\) | \(e\left(\frac{31}{88}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{29}{88}\right)\) | \(e\left(\frac{57}{88}\right)\) |
\(\chi_{3204}(1045,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{85}{88}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{35}{88}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{15}{88}\right)\) | \(e\left(\frac{37}{88}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{63}{88}\right)\) | \(e\left(\frac{51}{88}\right)\) |
\(\chi_{3204}(1081,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{15}{88}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{1}{88}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{13}{88}\right)\) | \(e\left(\frac{79}{88}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{37}{88}\right)\) | \(e\left(\frac{9}{88}\right)\) |
\(\chi_{3204}(1261,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{31}{88}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{49}{88}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{21}{88}\right)\) | \(e\left(\frac{87}{88}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{53}{88}\right)\) | \(e\left(\frac{1}{88}\right)\) |
\(\chi_{3204}(1297,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{39}{88}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{73}{88}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{69}{88}\right)\) | \(e\left(\frac{47}{88}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{61}{88}\right)\) | \(e\left(\frac{41}{88}\right)\) |
\(\chi_{3204}(1405,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{63}{88}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{57}{88}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{37}{88}\right)\) | \(e\left(\frac{15}{88}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{85}{88}\right)\) | \(e\left(\frac{73}{88}\right)\) |
\(\chi_{3204}(1621,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{19}{88}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{13}{88}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{81}{88}\right)\) | \(e\left(\frac{59}{88}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{41}{88}\right)\) | \(e\left(\frac{29}{88}\right)\) |
\(\chi_{3204}(1729,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{83}{88}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{29}{88}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{25}{88}\right)\) | \(e\left(\frac{3}{88}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{17}{88}\right)\) | \(e\left(\frac{85}{88}\right)\) |
\(\chi_{3204}(1765,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{75}{88}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{5}{88}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{65}{88}\right)\) | \(e\left(\frac{43}{88}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{9}{88}\right)\) | \(e\left(\frac{45}{88}\right)\) |
\(\chi_{3204}(1945,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{59}{88}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{45}{88}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{57}{88}\right)\) | \(e\left(\frac{35}{88}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{81}{88}\right)\) | \(e\left(\frac{53}{88}\right)\) |
\(\chi_{3204}(1981,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{41}{88}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{79}{88}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{59}{88}\right)\) | \(e\left(\frac{81}{88}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{19}{88}\right)\) | \(e\left(\frac{7}{88}\right)\) |
\(\chi_{3204}(2017,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{51}{88}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{21}{88}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{9}{88}\right)\) | \(e\left(\frac{75}{88}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{73}{88}\right)\) | \(e\left(\frac{13}{88}\right)\) |
\(\chi_{3204}(2053,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{57}{88}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{39}{88}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{67}{88}\right)\) | \(e\left(\frac{1}{88}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{35}{88}\right)\) | \(e\left(\frac{87}{88}\right)\) |
\(\chi_{3204}(2197,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{45}{88}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{3}{88}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{39}{88}\right)\) | \(e\left(\frac{61}{88}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{23}{88}\right)\) | \(e\left(\frac{27}{88}\right)\) |
\(\chi_{3204}(2341,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{67}{88}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{69}{88}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{17}{88}\right)\) | \(e\left(\frac{83}{88}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1}{88}\right)\) | \(e\left(\frac{5}{88}\right)\) |
\(\chi_{3204}(2377,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{35}{88}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{61}{88}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{1}{88}\right)\) | \(e\left(\frac{67}{88}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{57}{88}\right)\) | \(e\left(\frac{21}{88}\right)\) |
\(\chi_{3204}(2449,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{17}{88}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{7}{88}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{3}{88}\right)\) | \(e\left(\frac{25}{88}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{83}{88}\right)\) | \(e\left(\frac{63}{88}\right)\) |
\(\chi_{3204}(2485,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{5}{88}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{59}{88}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{63}{88}\right)\) | \(e\left(\frac{85}{88}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{71}{88}\right)\) | \(e\left(\frac{3}{88}\right)\) |