from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(137, base_ring=CyclotomicField(136))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,137))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(137\) | |
| Conductor: | \(137\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(136\) | 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: | $\Q(\zeta_{136})$ |
| Fixed field: | Number field defined by a degree 136 polynomial (not computed) |
First 31 of 64 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{137}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{1}{136}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{75}{136}\right)\) | \(e\left(\frac{11}{136}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{61}{68}\right)\) |
| \(\chi_{137}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{75}{136}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{49}{136}\right)\) | \(e\left(\frac{9}{136}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{19}{68}\right)\) |
| \(\chi_{137}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{11}{136}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{9}{136}\right)\) | \(e\left(\frac{121}{136}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{59}{68}\right)\) |
| \(\chi_{137}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{21}{136}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{79}{136}\right)\) | \(e\left(\frac{95}{136}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{57}{68}\right)\) |
| \(\chi_{137}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{25}{136}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{107}{136}\right)\) | \(e\left(\frac{3}{136}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{29}{68}\right)\) |
| \(\chi_{137}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{95}{136}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{53}{136}\right)\) | \(e\left(\frac{93}{136}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{15}{68}\right)\) |
| \(\chi_{137}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{43}{136}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{97}{136}\right)\) | \(e\left(\frac{65}{136}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{39}{68}\right)\) |
| \(\chi_{137}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{125}{136}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{127}{136}\right)\) | \(e\left(\frac{15}{136}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{68}\right)\) |
| \(\chi_{137}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{31}{136}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{13}{136}\right)\) | \(e\left(\frac{69}{136}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{55}{68}\right)\) |
| \(\chi_{137}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{35}{136}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{41}{136}\right)\) | \(e\left(\frac{113}{136}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{27}{68}\right)\) |
| \(\chi_{137}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{3}{136}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{89}{136}\right)\) | \(e\left(\frac{33}{136}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{47}{68}\right)\) |
| \(\chi_{137}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{91}{136}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{25}{136}\right)\) | \(e\left(\frac{49}{136}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{43}{68}\right)\) |
| \(\chi_{137}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{73}{136}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{35}{136}\right)\) | \(e\left(\frac{123}{136}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{33}{68}\right)\) |
| \(\chi_{137}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{123}{136}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{113}{136}\right)\) | \(e\left(\frac{129}{136}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{23}{68}\right)\) |
| \(\chi_{137}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{117}{136}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{71}{136}\right)\) | \(e\left(\frac{63}{136}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{65}{68}\right)\) |
| \(\chi_{137}(40,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{105}{136}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{123}{136}\right)\) | \(e\left(\frac{67}{136}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{68}\right)\) |
| \(\chi_{137}(42,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{53}{136}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{31}{136}\right)\) | \(e\left(\frac{39}{136}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{37}{68}\right)\) |
| \(\chi_{137}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{97}{136}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{67}{136}\right)\) | \(e\left(\frac{115}{136}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{68}\right)\) |
| \(\chi_{137}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{77}{136}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{63}{136}\right)\) | \(e\left(\frac{31}{136}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{68}\right)\) |
| \(\chi_{137}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{135}{136}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{61}{136}\right)\) | \(e\left(\frac{125}{136}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{68}\right)\) |
| \(\chi_{137}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{19}{136}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{65}{136}\right)\) | \(e\left(\frac{73}{136}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{68}\right)\) |
| \(\chi_{137}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{41}{136}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{83}{136}\right)\) | \(e\left(\frac{43}{136}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{53}{68}\right)\) |
| \(\chi_{137}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{39}{136}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{69}{136}\right)\) | \(e\left(\frac{21}{136}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{67}{68}\right)\) |
| \(\chi_{137}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{45}{136}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{111}{136}\right)\) | \(e\left(\frac{87}{136}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{25}{68}\right)\) |
| \(\chi_{137}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{131}{136}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{33}{136}\right)\) | \(e\left(\frac{81}{136}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{35}{68}\right)\) |
| \(\chi_{137}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{13}{136}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{23}{136}\right)\) | \(e\left(\frac{7}{136}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{45}{68}\right)\) |
| \(\chi_{137}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{61}{136}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{87}{136}\right)\) | \(e\left(\frac{127}{136}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{49}{68}\right)\) |
| \(\chi_{137}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{47}{136}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{125}{136}\right)\) | \(e\left(\frac{109}{136}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{68}\right)\) |
| \(\chi_{137}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{101}{136}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{95}{136}\right)\) | \(e\left(\frac{23}{136}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{41}{68}\right)\) |
| \(\chi_{137}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{83}{136}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{105}{136}\right)\) | \(e\left(\frac{97}{136}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{31}{68}\right)\) |
| \(\chi_{137}(66,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{133}{136}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{47}{136}\right)\) | \(e\left(\frac{103}{136}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{21}{68}\right)\) |