from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(223, base_ring=CyclotomicField(74))
M = H._module
chi = DirichletCharacter(H, M([49]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,223))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(223\) | |
| Conductor: | \(223\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(74\) | 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: | $\Q(\zeta_{37})$ |
| Fixed field: | Number field defined by a degree 74 polynomial |
First 31 of 36 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{223}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{37}\right)\) | \(e\left(\frac{49}{74}\right)\) | \(e\left(\frac{14}{37}\right)\) | \(e\left(\frac{69}{74}\right)\) | \(e\left(\frac{63}{74}\right)\) | \(e\left(\frac{2}{37}\right)\) | \(e\left(\frac{21}{37}\right)\) | \(e\left(\frac{12}{37}\right)\) | \(e\left(\frac{9}{74}\right)\) | \(e\left(\frac{63}{74}\right)\) |
| \(\chi_{223}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{37}\right)\) | \(e\left(\frac{35}{74}\right)\) | \(e\left(\frac{10}{37}\right)\) | \(e\left(\frac{7}{74}\right)\) | \(e\left(\frac{45}{74}\right)\) | \(e\left(\frac{12}{37}\right)\) | \(e\left(\frac{15}{37}\right)\) | \(e\left(\frac{35}{37}\right)\) | \(e\left(\frac{17}{74}\right)\) | \(e\left(\frac{45}{74}\right)\) |
| \(\chi_{223}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{37}\right)\) | \(e\left(\frac{1}{74}\right)\) | \(e\left(\frac{32}{37}\right)\) | \(e\left(\frac{15}{74}\right)\) | \(e\left(\frac{33}{74}\right)\) | \(e\left(\frac{31}{37}\right)\) | \(e\left(\frac{11}{37}\right)\) | \(e\left(\frac{1}{37}\right)\) | \(e\left(\frac{47}{74}\right)\) | \(e\left(\frac{33}{74}\right)\) |
| \(\chi_{223}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{37}\right)\) | \(e\left(\frac{21}{74}\right)\) | \(e\left(\frac{6}{37}\right)\) | \(e\left(\frac{19}{74}\right)\) | \(e\left(\frac{27}{74}\right)\) | \(e\left(\frac{22}{37}\right)\) | \(e\left(\frac{9}{37}\right)\) | \(e\left(\frac{21}{37}\right)\) | \(e\left(\frac{25}{74}\right)\) | \(e\left(\frac{27}{74}\right)\) |
| \(\chi_{223}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{37}\right)\) | \(e\left(\frac{61}{74}\right)\) | \(e\left(\frac{28}{37}\right)\) | \(e\left(\frac{27}{74}\right)\) | \(e\left(\frac{15}{74}\right)\) | \(e\left(\frac{4}{37}\right)\) | \(e\left(\frac{5}{37}\right)\) | \(e\left(\frac{24}{37}\right)\) | \(e\left(\frac{55}{74}\right)\) | \(e\left(\frac{15}{74}\right)\) |
| \(\chi_{223}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{37}\right)\) | \(e\left(\frac{3}{74}\right)\) | \(e\left(\frac{22}{37}\right)\) | \(e\left(\frac{45}{74}\right)\) | \(e\left(\frac{25}{74}\right)\) | \(e\left(\frac{19}{37}\right)\) | \(e\left(\frac{33}{37}\right)\) | \(e\left(\frac{3}{37}\right)\) | \(e\left(\frac{67}{74}\right)\) | \(e\left(\frac{25}{74}\right)\) |
| \(\chi_{223}(87,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{22}{37}\right)\) | \(e\left(\frac{43}{74}\right)\) | \(e\left(\frac{7}{37}\right)\) | \(e\left(\frac{53}{74}\right)\) | \(e\left(\frac{13}{74}\right)\) | \(e\left(\frac{1}{37}\right)\) | \(e\left(\frac{29}{37}\right)\) | \(e\left(\frac{6}{37}\right)\) | \(e\left(\frac{23}{74}\right)\) | \(e\left(\frac{13}{74}\right)\) |
| \(\chi_{223}(91,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{37}\right)\) | \(e\left(\frac{45}{74}\right)\) | \(e\left(\frac{34}{37}\right)\) | \(e\left(\frac{9}{74}\right)\) | \(e\left(\frac{5}{74}\right)\) | \(e\left(\frac{26}{37}\right)\) | \(e\left(\frac{14}{37}\right)\) | \(e\left(\frac{8}{37}\right)\) | \(e\left(\frac{43}{74}\right)\) | \(e\left(\frac{5}{74}\right)\) |
| \(\chi_{223}(95,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{37}\right)\) | \(e\left(\frac{13}{74}\right)\) | \(e\left(\frac{9}{37}\right)\) | \(e\left(\frac{47}{74}\right)\) | \(e\left(\frac{59}{74}\right)\) | \(e\left(\frac{33}{37}\right)\) | \(e\left(\frac{32}{37}\right)\) | \(e\left(\frac{13}{37}\right)\) | \(e\left(\frac{19}{74}\right)\) | \(e\left(\frac{59}{74}\right)\) |
| \(\chi_{223}(103,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{30}{37}\right)\) | \(e\left(\frac{25}{74}\right)\) | \(e\left(\frac{23}{37}\right)\) | \(e\left(\frac{5}{74}\right)\) | \(e\left(\frac{11}{74}\right)\) | \(e\left(\frac{35}{37}\right)\) | \(e\left(\frac{16}{37}\right)\) | \(e\left(\frac{25}{37}\right)\) | \(e\left(\frac{65}{74}\right)\) | \(e\left(\frac{11}{74}\right)\) |
| \(\chi_{223}(104,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{37}\right)\) | \(e\left(\frac{7}{74}\right)\) | \(e\left(\frac{2}{37}\right)\) | \(e\left(\frac{31}{74}\right)\) | \(e\left(\frac{9}{74}\right)\) | \(e\left(\frac{32}{37}\right)\) | \(e\left(\frac{3}{37}\right)\) | \(e\left(\frac{7}{37}\right)\) | \(e\left(\frac{33}{74}\right)\) | \(e\left(\frac{9}{74}\right)\) |
| \(\chi_{223}(108,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{12}{37}\right)\) | \(e\left(\frac{47}{74}\right)\) | \(e\left(\frac{24}{37}\right)\) | \(e\left(\frac{39}{74}\right)\) | \(e\left(\frac{71}{74}\right)\) | \(e\left(\frac{14}{37}\right)\) | \(e\left(\frac{36}{37}\right)\) | \(e\left(\frac{10}{37}\right)\) | \(e\left(\frac{63}{74}\right)\) | \(e\left(\frac{71}{74}\right)\) |
| \(\chi_{223}(111,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{37}\right)\) | \(e\left(\frac{51}{74}\right)\) | \(e\left(\frac{4}{37}\right)\) | \(e\left(\frac{25}{74}\right)\) | \(e\left(\frac{55}{74}\right)\) | \(e\left(\frac{27}{37}\right)\) | \(e\left(\frac{6}{37}\right)\) | \(e\left(\frac{14}{37}\right)\) | \(e\left(\frac{29}{74}\right)\) | \(e\left(\frac{55}{74}\right)\) |
| \(\chi_{223}(118,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{37}\right)\) | \(e\left(\frac{63}{74}\right)\) | \(e\left(\frac{18}{37}\right)\) | \(e\left(\frac{57}{74}\right)\) | \(e\left(\frac{7}{74}\right)\) | \(e\left(\frac{29}{37}\right)\) | \(e\left(\frac{27}{37}\right)\) | \(e\left(\frac{26}{37}\right)\) | \(e\left(\frac{1}{74}\right)\) | \(e\left(\frac{7}{74}\right)\) |
| \(\chi_{223}(125,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{18}{37}\right)\) | \(e\left(\frac{15}{74}\right)\) | \(e\left(\frac{36}{37}\right)\) | \(e\left(\frac{3}{74}\right)\) | \(e\left(\frac{51}{74}\right)\) | \(e\left(\frac{21}{37}\right)\) | \(e\left(\frac{17}{37}\right)\) | \(e\left(\frac{15}{37}\right)\) | \(e\left(\frac{39}{74}\right)\) | \(e\left(\frac{51}{74}\right)\) |
| \(\chi_{223}(141,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{37}\right)\) | \(e\left(\frac{17}{74}\right)\) | \(e\left(\frac{26}{37}\right)\) | \(e\left(\frac{33}{74}\right)\) | \(e\left(\frac{43}{74}\right)\) | \(e\left(\frac{9}{37}\right)\) | \(e\left(\frac{2}{37}\right)\) | \(e\left(\frac{17}{37}\right)\) | \(e\left(\frac{59}{74}\right)\) | \(e\left(\frac{43}{74}\right)\) |
| \(\chi_{223}(155,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{24}{37}\right)\) | \(e\left(\frac{57}{74}\right)\) | \(e\left(\frac{11}{37}\right)\) | \(e\left(\frac{41}{74}\right)\) | \(e\left(\frac{31}{74}\right)\) | \(e\left(\frac{28}{37}\right)\) | \(e\left(\frac{35}{37}\right)\) | \(e\left(\frac{20}{37}\right)\) | \(e\left(\frac{15}{74}\right)\) | \(e\left(\frac{31}{74}\right)\) |
| \(\chi_{223}(157,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{37}\right)\) | \(e\left(\frac{59}{74}\right)\) | \(e\left(\frac{1}{37}\right)\) | \(e\left(\frac{71}{74}\right)\) | \(e\left(\frac{23}{74}\right)\) | \(e\left(\frac{16}{37}\right)\) | \(e\left(\frac{20}{37}\right)\) | \(e\left(\frac{22}{37}\right)\) | \(e\left(\frac{35}{74}\right)\) | \(e\left(\frac{23}{74}\right)\) |
| \(\chi_{223}(159,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{37}\right)\) | \(e\left(\frac{27}{74}\right)\) | \(e\left(\frac{13}{37}\right)\) | \(e\left(\frac{35}{74}\right)\) | \(e\left(\frac{3}{74}\right)\) | \(e\left(\frac{23}{37}\right)\) | \(e\left(\frac{1}{37}\right)\) | \(e\left(\frac{27}{37}\right)\) | \(e\left(\frac{11}{74}\right)\) | \(e\left(\frac{3}{74}\right)\) |
| \(\chi_{223}(163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{32}{37}\right)\) | \(e\left(\frac{39}{74}\right)\) | \(e\left(\frac{27}{37}\right)\) | \(e\left(\frac{67}{74}\right)\) | \(e\left(\frac{29}{74}\right)\) | \(e\left(\frac{25}{37}\right)\) | \(e\left(\frac{22}{37}\right)\) | \(e\left(\frac{2}{37}\right)\) | \(e\left(\frac{57}{74}\right)\) | \(e\left(\frac{29}{74}\right)\) |
| \(\chi_{223}(167,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{37}\right)\) | \(e\left(\frac{65}{74}\right)\) | \(e\left(\frac{8}{37}\right)\) | \(e\left(\frac{13}{74}\right)\) | \(e\left(\frac{73}{74}\right)\) | \(e\left(\frac{17}{37}\right)\) | \(e\left(\frac{12}{37}\right)\) | \(e\left(\frac{28}{37}\right)\) | \(e\left(\frac{21}{74}\right)\) | \(e\left(\frac{73}{74}\right)\) |
| \(\chi_{223}(174,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{20}{37}\right)\) | \(e\left(\frac{29}{74}\right)\) | \(e\left(\frac{3}{37}\right)\) | \(e\left(\frac{65}{74}\right)\) | \(e\left(\frac{69}{74}\right)\) | \(e\left(\frac{11}{37}\right)\) | \(e\left(\frac{23}{37}\right)\) | \(e\left(\frac{29}{37}\right)\) | \(e\left(\frac{31}{74}\right)\) | \(e\left(\frac{69}{74}\right)\) |
| \(\chi_{223}(182,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{37}\right)\) | \(e\left(\frac{31}{74}\right)\) | \(e\left(\frac{30}{37}\right)\) | \(e\left(\frac{21}{74}\right)\) | \(e\left(\frac{61}{74}\right)\) | \(e\left(\frac{36}{37}\right)\) | \(e\left(\frac{8}{37}\right)\) | \(e\left(\frac{31}{37}\right)\) | \(e\left(\frac{51}{74}\right)\) | \(e\left(\frac{61}{74}\right)\) |
| \(\chi_{223}(189,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{26}{37}\right)\) | \(e\left(\frac{71}{74}\right)\) | \(e\left(\frac{15}{37}\right)\) | \(e\left(\frac{29}{74}\right)\) | \(e\left(\frac{49}{74}\right)\) | \(e\left(\frac{18}{37}\right)\) | \(e\left(\frac{4}{37}\right)\) | \(e\left(\frac{34}{37}\right)\) | \(e\left(\frac{7}{74}\right)\) | \(e\left(\frac{49}{74}\right)\) |
| \(\chi_{223}(190,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{37}\right)\) | \(e\left(\frac{73}{74}\right)\) | \(e\left(\frac{5}{37}\right)\) | \(e\left(\frac{59}{74}\right)\) | \(e\left(\frac{41}{74}\right)\) | \(e\left(\frac{6}{37}\right)\) | \(e\left(\frac{26}{37}\right)\) | \(e\left(\frac{36}{37}\right)\) | \(e\left(\frac{27}{74}\right)\) | \(e\left(\frac{41}{74}\right)\) |
| \(\chi_{223}(191,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{37}\right)\) | \(e\left(\frac{41}{74}\right)\) | \(e\left(\frac{17}{37}\right)\) | \(e\left(\frac{23}{74}\right)\) | \(e\left(\frac{21}{74}\right)\) | \(e\left(\frac{13}{37}\right)\) | \(e\left(\frac{7}{37}\right)\) | \(e\left(\frac{4}{37}\right)\) | \(e\left(\frac{3}{74}\right)\) | \(e\left(\frac{21}{74}\right)\) |
| \(\chi_{223}(193,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{34}{37}\right)\) | \(e\left(\frac{53}{74}\right)\) | \(e\left(\frac{31}{37}\right)\) | \(e\left(\frac{55}{74}\right)\) | \(e\left(\frac{47}{74}\right)\) | \(e\left(\frac{15}{37}\right)\) | \(e\left(\frac{28}{37}\right)\) | \(e\left(\frac{16}{37}\right)\) | \(e\left(\frac{49}{74}\right)\) | \(e\left(\frac{47}{74}\right)\) |
| \(\chi_{223}(195,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{37}\right)\) | \(e\left(\frac{5}{74}\right)\) | \(e\left(\frac{12}{37}\right)\) | \(e\left(\frac{1}{74}\right)\) | \(e\left(\frac{17}{74}\right)\) | \(e\left(\frac{7}{37}\right)\) | \(e\left(\frac{18}{37}\right)\) | \(e\left(\frac{5}{37}\right)\) | \(e\left(\frac{13}{74}\right)\) | \(e\left(\frac{17}{74}\right)\) |
| \(\chi_{223}(206,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{28}{37}\right)\) | \(e\left(\frac{11}{74}\right)\) | \(e\left(\frac{19}{37}\right)\) | \(e\left(\frac{17}{74}\right)\) | \(e\left(\frac{67}{74}\right)\) | \(e\left(\frac{8}{37}\right)\) | \(e\left(\frac{10}{37}\right)\) | \(e\left(\frac{11}{37}\right)\) | \(e\left(\frac{73}{74}\right)\) | \(e\left(\frac{67}{74}\right)\) |
| \(\chi_{223}(207,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{37}\right)\) | \(e\left(\frac{55}{74}\right)\) | \(e\left(\frac{21}{37}\right)\) | \(e\left(\frac{11}{74}\right)\) | \(e\left(\frac{39}{74}\right)\) | \(e\left(\frac{3}{37}\right)\) | \(e\left(\frac{13}{37}\right)\) | \(e\left(\frac{18}{37}\right)\) | \(e\left(\frac{69}{74}\right)\) | \(e\left(\frac{39}{74}\right)\) |
| \(\chi_{223}(208,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{36}{37}\right)\) | \(e\left(\frac{67}{74}\right)\) | \(e\left(\frac{35}{37}\right)\) | \(e\left(\frac{43}{74}\right)\) | \(e\left(\frac{65}{74}\right)\) | \(e\left(\frac{5}{37}\right)\) | \(e\left(\frac{34}{37}\right)\) | \(e\left(\frac{30}{37}\right)\) | \(e\left(\frac{41}{74}\right)\) | \(e\left(\frac{65}{74}\right)\) |