from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(107, base_ring=CyclotomicField(106))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,107))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(107\) | |
| Conductor: | \(107\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(106\) | 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: | $\Q(\zeta_{53})$ |
| Fixed field: | Number field defined by a degree 106 polynomial (not computed) |
First 31 of 52 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{107}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{106}\right)\) | \(e\left(\frac{35}{53}\right)\) | \(e\left(\frac{1}{53}\right)\) | \(e\left(\frac{47}{106}\right)\) | \(e\left(\frac{71}{106}\right)\) | \(e\left(\frac{43}{106}\right)\) | \(e\left(\frac{3}{106}\right)\) | \(e\left(\frac{17}{53}\right)\) | \(e\left(\frac{24}{53}\right)\) | \(e\left(\frac{11}{53}\right)\) |
| \(\chi_{107}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{106}\right)\) | \(e\left(\frac{2}{53}\right)\) | \(e\left(\frac{47}{53}\right)\) | \(e\left(\frac{89}{106}\right)\) | \(e\left(\frac{51}{106}\right)\) | \(e\left(\frac{7}{106}\right)\) | \(e\left(\frac{35}{106}\right)\) | \(e\left(\frac{4}{53}\right)\) | \(e\left(\frac{15}{53}\right)\) | \(e\left(\frac{40}{53}\right)\) |
| \(\chi_{107}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{106}\right)\) | \(e\left(\frac{47}{53}\right)\) | \(e\left(\frac{18}{53}\right)\) | \(e\left(\frac{51}{106}\right)\) | \(e\left(\frac{59}{106}\right)\) | \(e\left(\frac{85}{106}\right)\) | \(e\left(\frac{1}{106}\right)\) | \(e\left(\frac{41}{53}\right)\) | \(e\left(\frac{8}{53}\right)\) | \(e\left(\frac{39}{53}\right)\) |
| \(\chi_{107}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{106}\right)\) | \(e\left(\frac{21}{53}\right)\) | \(e\left(\frac{43}{53}\right)\) | \(e\left(\frac{7}{106}\right)\) | \(e\left(\frac{85}{106}\right)\) | \(e\left(\frac{47}{106}\right)\) | \(e\left(\frac{23}{106}\right)\) | \(e\left(\frac{42}{53}\right)\) | \(e\left(\frac{25}{53}\right)\) | \(e\left(\frac{49}{53}\right)\) |
| \(\chi_{107}(8,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{106}\right)\) | \(e\left(\frac{52}{53}\right)\) | \(e\left(\frac{3}{53}\right)\) | \(e\left(\frac{35}{106}\right)\) | \(e\left(\frac{1}{106}\right)\) | \(e\left(\frac{23}{106}\right)\) | \(e\left(\frac{9}{106}\right)\) | \(e\left(\frac{51}{53}\right)\) | \(e\left(\frac{19}{53}\right)\) | \(e\left(\frac{33}{53}\right)\) |
| \(\chi_{107}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{106}\right)\) | \(e\left(\frac{14}{53}\right)\) | \(e\left(\frac{11}{53}\right)\) | \(e\left(\frac{93}{106}\right)\) | \(e\left(\frac{39}{106}\right)\) | \(e\left(\frac{49}{106}\right)\) | \(e\left(\frac{33}{106}\right)\) | \(e\left(\frac{28}{53}\right)\) | \(e\left(\frac{52}{53}\right)\) | \(e\left(\frac{15}{53}\right)\) |
| \(\chi_{107}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{106}\right)\) | \(e\left(\frac{8}{53}\right)\) | \(e\left(\frac{29}{53}\right)\) | \(e\left(\frac{91}{106}\right)\) | \(e\left(\frac{45}{106}\right)\) | \(e\left(\frac{81}{106}\right)\) | \(e\left(\frac{87}{106}\right)\) | \(e\left(\frac{16}{53}\right)\) | \(e\left(\frac{7}{53}\right)\) | \(e\left(\frac{1}{53}\right)\) |
| \(\chi_{107}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{106}\right)\) | \(e\left(\frac{6}{53}\right)\) | \(e\left(\frac{35}{53}\right)\) | \(e\left(\frac{55}{106}\right)\) | \(e\left(\frac{47}{106}\right)\) | \(e\left(\frac{21}{106}\right)\) | \(e\left(\frac{105}{106}\right)\) | \(e\left(\frac{12}{53}\right)\) | \(e\left(\frac{45}{53}\right)\) | \(e\left(\frac{14}{53}\right)\) |
| \(\chi_{107}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{106}\right)\) | \(e\left(\frac{19}{53}\right)\) | \(e\left(\frac{49}{53}\right)\) | \(e\left(\frac{77}{106}\right)\) | \(e\left(\frac{87}{106}\right)\) | \(e\left(\frac{93}{106}\right)\) | \(e\left(\frac{41}{106}\right)\) | \(e\left(\frac{38}{53}\right)\) | \(e\left(\frac{10}{53}\right)\) | \(e\left(\frac{9}{53}\right)\) |
| \(\chi_{107}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{106}\right)\) | \(e\left(\frac{33}{53}\right)\) | \(e\left(\frac{7}{53}\right)\) | \(e\left(\frac{11}{106}\right)\) | \(e\left(\frac{73}{106}\right)\) | \(e\left(\frac{89}{106}\right)\) | \(e\left(\frac{21}{106}\right)\) | \(e\left(\frac{13}{53}\right)\) | \(e\left(\frac{9}{53}\right)\) | \(e\left(\frac{24}{53}\right)\) |
| \(\chi_{107}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{106}\right)\) | \(e\left(\frac{10}{53}\right)\) | \(e\left(\frac{23}{53}\right)\) | \(e\left(\frac{21}{106}\right)\) | \(e\left(\frac{43}{106}\right)\) | \(e\left(\frac{35}{106}\right)\) | \(e\left(\frac{69}{106}\right)\) | \(e\left(\frac{20}{53}\right)\) | \(e\left(\frac{22}{53}\right)\) | \(e\left(\frac{41}{53}\right)\) |
| \(\chi_{107}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{106}\right)\) | \(e\left(\frac{11}{53}\right)\) | \(e\left(\frac{20}{53}\right)\) | \(e\left(\frac{39}{106}\right)\) | \(e\left(\frac{95}{106}\right)\) | \(e\left(\frac{65}{106}\right)\) | \(e\left(\frac{7}{106}\right)\) | \(e\left(\frac{22}{53}\right)\) | \(e\left(\frac{3}{53}\right)\) | \(e\left(\frac{8}{53}\right)\) |
| \(\chi_{107}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{106}\right)\) | \(e\left(\frac{48}{53}\right)\) | \(e\left(\frac{15}{53}\right)\) | \(e\left(\frac{69}{106}\right)\) | \(e\left(\frac{5}{106}\right)\) | \(e\left(\frac{9}{106}\right)\) | \(e\left(\frac{45}{106}\right)\) | \(e\left(\frac{43}{53}\right)\) | \(e\left(\frac{42}{53}\right)\) | \(e\left(\frac{6}{53}\right)\) |
| \(\chi_{107}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{106}\right)\) | \(e\left(\frac{38}{53}\right)\) | \(e\left(\frac{45}{53}\right)\) | \(e\left(\frac{101}{106}\right)\) | \(e\left(\frac{15}{106}\right)\) | \(e\left(\frac{27}{106}\right)\) | \(e\left(\frac{29}{106}\right)\) | \(e\left(\frac{23}{53}\right)\) | \(e\left(\frac{20}{53}\right)\) | \(e\left(\frac{18}{53}\right)\) |
| \(\chi_{107}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{106}\right)\) | \(e\left(\frac{44}{53}\right)\) | \(e\left(\frac{27}{53}\right)\) | \(e\left(\frac{103}{106}\right)\) | \(e\left(\frac{9}{106}\right)\) | \(e\left(\frac{101}{106}\right)\) | \(e\left(\frac{81}{106}\right)\) | \(e\left(\frac{35}{53}\right)\) | \(e\left(\frac{12}{53}\right)\) | \(e\left(\frac{32}{53}\right)\) |
| \(\chi_{107}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{106}\right)\) | \(e\left(\frac{16}{53}\right)\) | \(e\left(\frac{5}{53}\right)\) | \(e\left(\frac{23}{106}\right)\) | \(e\left(\frac{37}{106}\right)\) | \(e\left(\frac{3}{106}\right)\) | \(e\left(\frac{15}{106}\right)\) | \(e\left(\frac{32}{53}\right)\) | \(e\left(\frac{14}{53}\right)\) | \(e\left(\frac{2}{53}\right)\) |
| \(\chi_{107}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{106}\right)\) | \(e\left(\frac{9}{53}\right)\) | \(e\left(\frac{26}{53}\right)\) | \(e\left(\frac{3}{106}\right)\) | \(e\left(\frac{97}{106}\right)\) | \(e\left(\frac{5}{106}\right)\) | \(e\left(\frac{25}{106}\right)\) | \(e\left(\frac{18}{53}\right)\) | \(e\left(\frac{41}{53}\right)\) | \(e\left(\frac{21}{53}\right)\) |
| \(\chi_{107}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{106}\right)\) | \(e\left(\frac{51}{53}\right)\) | \(e\left(\frac{6}{53}\right)\) | \(e\left(\frac{17}{106}\right)\) | \(e\left(\frac{55}{106}\right)\) | \(e\left(\frac{99}{106}\right)\) | \(e\left(\frac{71}{106}\right)\) | \(e\left(\frac{49}{53}\right)\) | \(e\left(\frac{38}{53}\right)\) | \(e\left(\frac{13}{53}\right)\) |
| \(\chi_{107}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{81}{106}\right)\) | \(e\left(\frac{26}{53}\right)\) | \(e\left(\frac{28}{53}\right)\) | \(e\left(\frac{97}{106}\right)\) | \(e\left(\frac{27}{106}\right)\) | \(e\left(\frac{91}{106}\right)\) | \(e\left(\frac{31}{106}\right)\) | \(e\left(\frac{52}{53}\right)\) | \(e\left(\frac{36}{53}\right)\) | \(e\left(\frac{43}{53}\right)\) |
| \(\chi_{107}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{106}\right)\) | \(e\left(\frac{32}{53}\right)\) | \(e\left(\frac{10}{53}\right)\) | \(e\left(\frac{99}{106}\right)\) | \(e\left(\frac{21}{106}\right)\) | \(e\left(\frac{59}{106}\right)\) | \(e\left(\frac{83}{106}\right)\) | \(e\left(\frac{11}{53}\right)\) | \(e\left(\frac{28}{53}\right)\) | \(e\left(\frac{4}{53}\right)\) |
| \(\chi_{107}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{106}\right)\) | \(e\left(\frac{39}{53}\right)\) | \(e\left(\frac{42}{53}\right)\) | \(e\left(\frac{13}{106}\right)\) | \(e\left(\frac{67}{106}\right)\) | \(e\left(\frac{57}{106}\right)\) | \(e\left(\frac{73}{106}\right)\) | \(e\left(\frac{25}{53}\right)\) | \(e\left(\frac{1}{53}\right)\) | \(e\left(\frac{38}{53}\right)\) |
| \(\chi_{107}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{99}{106}\right)\) | \(e\left(\frac{20}{53}\right)\) | \(e\left(\frac{46}{53}\right)\) | \(e\left(\frac{95}{106}\right)\) | \(e\left(\frac{33}{106}\right)\) | \(e\left(\frac{17}{106}\right)\) | \(e\left(\frac{85}{106}\right)\) | \(e\left(\frac{40}{53}\right)\) | \(e\left(\frac{44}{53}\right)\) | \(e\left(\frac{29}{53}\right)\) |
| \(\chi_{107}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{105}{106}\right)\) | \(e\left(\frac{18}{53}\right)\) | \(e\left(\frac{52}{53}\right)\) | \(e\left(\frac{59}{106}\right)\) | \(e\left(\frac{35}{106}\right)\) | \(e\left(\frac{63}{106}\right)\) | \(e\left(\frac{103}{106}\right)\) | \(e\left(\frac{36}{53}\right)\) | \(e\left(\frac{29}{53}\right)\) | \(e\left(\frac{42}{53}\right)\) |
| \(\chi_{107}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{106}\right)\) | \(e\left(\frac{30}{53}\right)\) | \(e\left(\frac{16}{53}\right)\) | \(e\left(\frac{63}{106}\right)\) | \(e\left(\frac{23}{106}\right)\) | \(e\left(\frac{105}{106}\right)\) | \(e\left(\frac{101}{106}\right)\) | \(e\left(\frac{7}{53}\right)\) | \(e\left(\frac{13}{53}\right)\) | \(e\left(\frac{17}{53}\right)\) |
| \(\chi_{107}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{106}\right)\) | \(e\left(\frac{42}{53}\right)\) | \(e\left(\frac{33}{53}\right)\) | \(e\left(\frac{67}{106}\right)\) | \(e\left(\frac{11}{106}\right)\) | \(e\left(\frac{41}{106}\right)\) | \(e\left(\frac{99}{106}\right)\) | \(e\left(\frac{31}{53}\right)\) | \(e\left(\frac{50}{53}\right)\) | \(e\left(\frac{45}{53}\right)\) |
| \(\chi_{107}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{106}\right)\) | \(e\left(\frac{46}{53}\right)\) | \(e\left(\frac{21}{53}\right)\) | \(e\left(\frac{33}{106}\right)\) | \(e\left(\frac{7}{106}\right)\) | \(e\left(\frac{55}{106}\right)\) | \(e\left(\frac{63}{106}\right)\) | \(e\left(\frac{39}{53}\right)\) | \(e\left(\frac{27}{53}\right)\) | \(e\left(\frac{19}{53}\right)\) |
| \(\chi_{107}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{106}\right)\) | \(e\left(\frac{31}{53}\right)\) | \(e\left(\frac{13}{53}\right)\) | \(e\left(\frac{81}{106}\right)\) | \(e\left(\frac{75}{106}\right)\) | \(e\left(\frac{29}{106}\right)\) | \(e\left(\frac{39}{106}\right)\) | \(e\left(\frac{9}{53}\right)\) | \(e\left(\frac{47}{53}\right)\) | \(e\left(\frac{37}{53}\right)\) |
| \(\chi_{107}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{106}\right)\) | \(e\left(\frac{45}{53}\right)\) | \(e\left(\frac{24}{53}\right)\) | \(e\left(\frac{15}{106}\right)\) | \(e\left(\frac{61}{106}\right)\) | \(e\left(\frac{25}{106}\right)\) | \(e\left(\frac{19}{106}\right)\) | \(e\left(\frac{37}{53}\right)\) | \(e\left(\frac{46}{53}\right)\) | \(e\left(\frac{52}{53}\right)\) |
| \(\chi_{107}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{106}\right)\) | \(e\left(\frac{15}{53}\right)\) | \(e\left(\frac{8}{53}\right)\) | \(e\left(\frac{5}{106}\right)\) | \(e\left(\frac{91}{106}\right)\) | \(e\left(\frac{79}{106}\right)\) | \(e\left(\frac{77}{106}\right)\) | \(e\left(\frac{30}{53}\right)\) | \(e\left(\frac{33}{53}\right)\) | \(e\left(\frac{35}{53}\right)\) |
| \(\chi_{107}(66,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{93}{106}\right)\) | \(e\left(\frac{22}{53}\right)\) | \(e\left(\frac{40}{53}\right)\) | \(e\left(\frac{25}{106}\right)\) | \(e\left(\frac{31}{106}\right)\) | \(e\left(\frac{77}{106}\right)\) | \(e\left(\frac{67}{106}\right)\) | \(e\left(\frac{44}{53}\right)\) | \(e\left(\frac{6}{53}\right)\) | \(e\left(\frac{16}{53}\right)\) |
| \(\chi_{107}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{106}\right)\) | \(e\left(\frac{1}{53}\right)\) | \(e\left(\frac{50}{53}\right)\) | \(e\left(\frac{71}{106}\right)\) | \(e\left(\frac{105}{106}\right)\) | \(e\left(\frac{83}{106}\right)\) | \(e\left(\frac{97}{106}\right)\) | \(e\left(\frac{2}{53}\right)\) | \(e\left(\frac{34}{53}\right)\) | \(e\left(\frac{20}{53}\right)\) |