from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(197, base_ring=CyclotomicField(196))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,197))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(197\) | |
| Conductor: | \(197\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(196\) | 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: | $\Q(\zeta_{196})$ |
| Fixed field: | Number field defined by a degree 196 polynomial (not computed) |
First 31 of 84 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{197}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{196}\right)\) | \(e\left(\frac{181}{196}\right)\) | \(e\left(\frac{1}{98}\right)\) | \(e\left(\frac{89}{196}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{73}{98}\right)\) | \(e\left(\frac{3}{196}\right)\) | \(e\left(\frac{83}{98}\right)\) | \(e\left(\frac{45}{98}\right)\) | \(e\left(\frac{29}{196}\right)\) |
| \(\chi_{197}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{196}\right)\) | \(e\left(\frac{29}{196}\right)\) | \(e\left(\frac{83}{98}\right)\) | \(e\left(\frac{37}{196}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{81}{98}\right)\) | \(e\left(\frac{151}{196}\right)\) | \(e\left(\frac{29}{98}\right)\) | \(e\left(\frac{11}{98}\right)\) | \(e\left(\frac{153}{196}\right)\) |
| \(\chi_{197}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{196}\right)\) | \(e\left(\frac{37}{196}\right)\) | \(e\left(\frac{89}{98}\right)\) | \(e\left(\frac{81}{196}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{29}{98}\right)\) | \(e\left(\frac{71}{196}\right)\) | \(e\left(\frac{37}{98}\right)\) | \(e\left(\frac{85}{98}\right)\) | \(e\left(\frac{33}{196}\right)\) |
| \(\chi_{197}(8,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{196}\right)\) | \(e\left(\frac{151}{196}\right)\) | \(e\left(\frac{3}{98}\right)\) | \(e\left(\frac{71}{196}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{23}{98}\right)\) | \(e\left(\frac{9}{196}\right)\) | \(e\left(\frac{53}{98}\right)\) | \(e\left(\frac{37}{98}\right)\) | \(e\left(\frac{87}{196}\right)\) |
| \(\chi_{197}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{196}\right)\) | \(e\left(\frac{153}{196}\right)\) | \(e\left(\frac{29}{98}\right)\) | \(e\left(\frac{33}{196}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{59}{98}\right)\) | \(e\left(\frac{87}{196}\right)\) | \(e\left(\frac{55}{98}\right)\) | \(e\left(\frac{31}{98}\right)\) | \(e\left(\frac{57}{196}\right)\) |
| \(\chi_{197}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{183}{196}\right)\) | \(e\left(\frac{195}{196}\right)\) | \(e\left(\frac{85}{98}\right)\) | \(e\left(\frac{19}{196}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{31}{98}\right)\) | \(e\left(\frac{157}{196}\right)\) | \(e\left(\frac{97}{98}\right)\) | \(e\left(\frac{3}{98}\right)\) | \(e\left(\frac{15}{196}\right)\) |
| \(\chi_{197}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{196}\right)\) | \(e\left(\frac{17}{196}\right)\) | \(e\left(\frac{25}{98}\right)\) | \(e\left(\frac{69}{196}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{61}{98}\right)\) | \(e\left(\frac{75}{196}\right)\) | \(e\left(\frac{17}{98}\right)\) | \(e\left(\frac{47}{98}\right)\) | \(e\left(\frac{137}{196}\right)\) |
| \(\chi_{197}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{159}{196}\right)\) | \(e\left(\frac{163}{196}\right)\) | \(e\left(\frac{61}{98}\right)\) | \(e\left(\frac{39}{196}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{43}{98}\right)\) | \(e\left(\frac{85}{196}\right)\) | \(e\left(\frac{65}{98}\right)\) | \(e\left(\frac{1}{98}\right)\) | \(e\left(\frac{103}{196}\right)\) |
| \(\chi_{197}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{196}\right)\) | \(e\left(\frac{43}{196}\right)\) | \(e\left(\frac{69}{98}\right)\) | \(e\left(\frac{163}{196}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{39}{98}\right)\) | \(e\left(\frac{109}{196}\right)\) | \(e\left(\frac{43}{98}\right)\) | \(e\left(\frac{67}{98}\right)\) | \(e\left(\frac{139}{196}\right)\) |
| \(\chi_{197}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{196}\right)\) | \(e\left(\frac{191}{196}\right)\) | \(e\left(\frac{33}{98}\right)\) | \(e\left(\frac{95}{196}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{57}{98}\right)\) | \(e\left(\frac{1}{196}\right)\) | \(e\left(\frac{93}{98}\right)\) | \(e\left(\frac{15}{98}\right)\) | \(e\left(\frac{75}{196}\right)\) |
| \(\chi_{197}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{196}\right)\) | \(e\left(\frac{87}{196}\right)\) | \(e\left(\frac{53}{98}\right)\) | \(e\left(\frac{111}{196}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{47}{98}\right)\) | \(e\left(\frac{61}{196}\right)\) | \(e\left(\frac{87}{98}\right)\) | \(e\left(\frac{33}{98}\right)\) | \(e\left(\frac{67}{196}\right)\) |
| \(\chi_{197}(30,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{75}{196}\right)\) | \(e\left(\frac{51}{196}\right)\) | \(e\left(\frac{75}{98}\right)\) | \(e\left(\frac{11}{196}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{85}{98}\right)\) | \(e\left(\frac{29}{196}\right)\) | \(e\left(\frac{51}{98}\right)\) | \(e\left(\frac{43}{98}\right)\) | \(e\left(\frac{19}{196}\right)\) |
| \(\chi_{197}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{141}{196}\right)\) | \(e\left(\frac{41}{196}\right)\) | \(e\left(\frac{43}{98}\right)\) | \(e\left(\frac{5}{196}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{3}{98}\right)\) | \(e\left(\frac{31}{196}\right)\) | \(e\left(\frac{41}{98}\right)\) | \(e\left(\frac{73}{98}\right)\) | \(e\left(\frac{169}{196}\right)\) |
| \(\chi_{197}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{196}\right)\) | \(e\left(\frac{121}{196}\right)\) | \(e\left(\frac{5}{98}\right)\) | \(e\left(\frac{53}{196}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{71}{98}\right)\) | \(e\left(\frac{15}{196}\right)\) | \(e\left(\frac{23}{98}\right)\) | \(e\left(\frac{29}{98}\right)\) | \(e\left(\frac{145}{196}\right)\) |
| \(\chi_{197}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{196}\right)\) | \(e\left(\frac{3}{196}\right)\) | \(e\left(\frac{39}{98}\right)\) | \(e\left(\frac{139}{196}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{5}{98}\right)\) | \(e\left(\frac{117}{196}\right)\) | \(e\left(\frac{3}{98}\right)\) | \(e\left(\frac{89}{98}\right)\) | \(e\left(\frac{151}{196}\right)\) |
| \(\chi_{197}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{155}{196}\right)\) | \(e\left(\frac{27}{196}\right)\) | \(e\left(\frac{57}{98}\right)\) | \(e\left(\frac{75}{196}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{45}{98}\right)\) | \(e\left(\frac{73}{196}\right)\) | \(e\left(\frac{27}{98}\right)\) | \(e\left(\frac{17}{98}\right)\) | \(e\left(\frac{183}{196}\right)\) |
| \(\chi_{197}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{196}\right)\) | \(e\left(\frac{123}{196}\right)\) | \(e\left(\frac{31}{98}\right)\) | \(e\left(\frac{15}{196}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{9}{98}\right)\) | \(e\left(\frac{93}{196}\right)\) | \(e\left(\frac{25}{98}\right)\) | \(e\left(\frac{23}{98}\right)\) | \(e\left(\frac{115}{196}\right)\) |
| \(\chi_{197}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{196}\right)\) | \(e\left(\frac{95}{196}\right)\) | \(e\left(\frac{59}{98}\right)\) | \(e\left(\frac{155}{196}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{93}{98}\right)\) | \(e\left(\frac{177}{196}\right)\) | \(e\left(\frac{95}{98}\right)\) | \(e\left(\frac{9}{98}\right)\) | \(e\left(\frac{143}{196}\right)\) |
| \(\chi_{197}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{196}\right)\) | \(e\left(\frac{145}{196}\right)\) | \(e\left(\frac{23}{98}\right)\) | \(e\left(\frac{185}{196}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{13}{98}\right)\) | \(e\left(\frac{167}{196}\right)\) | \(e\left(\frac{47}{98}\right)\) | \(e\left(\frac{55}{98}\right)\) | \(e\left(\frac{177}{196}\right)\) |
| \(\chi_{197}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{185}{196}\right)\) | \(e\left(\frac{165}{196}\right)\) | \(e\left(\frac{87}{98}\right)\) | \(e\left(\frac{1}{196}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{79}{98}\right)\) | \(e\left(\frac{163}{196}\right)\) | \(e\left(\frac{67}{98}\right)\) | \(e\left(\frac{93}{98}\right)\) | \(e\left(\frac{73}{196}\right)\) |
| \(\chi_{197}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{196}\right)\) | \(e\left(\frac{59}{196}\right)\) | \(e\left(\frac{81}{98}\right)\) | \(e\left(\frac{55}{196}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{33}{98}\right)\) | \(e\left(\frac{145}{196}\right)\) | \(e\left(\frac{59}{98}\right)\) | \(e\left(\frac{19}{98}\right)\) | \(e\left(\frac{95}{196}\right)\) |
| \(\chi_{197}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{196}\right)\) | \(e\left(\frac{183}{196}\right)\) | \(e\left(\frac{27}{98}\right)\) | \(e\left(\frac{51}{196}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{11}{98}\right)\) | \(e\left(\frac{81}{196}\right)\) | \(e\left(\frac{85}{98}\right)\) | \(e\left(\frac{39}{98}\right)\) | \(e\left(\frac{195}{196}\right)\) |
| \(\chi_{197}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{196}\right)\) | \(e\left(\frac{117}{196}\right)\) | \(e\left(\frac{51}{98}\right)\) | \(e\left(\frac{129}{196}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{97}{98}\right)\) | \(e\left(\frac{55}{196}\right)\) | \(e\left(\frac{19}{98}\right)\) | \(e\left(\frac{41}{98}\right)\) | \(e\left(\frac{9}{196}\right)\) |
| \(\chi_{197}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{196}\right)\) | \(e\left(\frac{71}{196}\right)\) | \(e\left(\frac{41}{98}\right)\) | \(e\left(\frac{23}{196}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{53}{98}\right)\) | \(e\left(\frac{25}{196}\right)\) | \(e\left(\frac{71}{98}\right)\) | \(e\left(\frac{81}{98}\right)\) | \(e\left(\frac{111}{196}\right)\) |
| \(\chi_{197}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{196}\right)\) | \(e\left(\frac{33}{196}\right)\) | \(e\left(\frac{37}{98}\right)\) | \(e\left(\frac{157}{196}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{55}{98}\right)\) | \(e\left(\frac{111}{196}\right)\) | \(e\left(\frac{33}{98}\right)\) | \(e\left(\frac{97}{98}\right)\) | \(e\left(\frac{93}{196}\right)\) |
| \(\chi_{197}(66,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{196}\right)\) | \(e\left(\frac{167}{196}\right)\) | \(e\left(\frac{15}{98}\right)\) | \(e\left(\frac{159}{196}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{17}{98}\right)\) | \(e\left(\frac{45}{196}\right)\) | \(e\left(\frac{69}{98}\right)\) | \(e\left(\frac{87}{98}\right)\) | \(e\left(\frac{43}{196}\right)\) |
| \(\chi_{197}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{196}\right)\) | \(e\left(\frac{137}{196}\right)\) | \(e\left(\frac{17}{98}\right)\) | \(e\left(\frac{141}{196}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{65}{98}\right)\) | \(e\left(\frac{51}{196}\right)\) | \(e\left(\frac{39}{98}\right)\) | \(e\left(\frac{79}{98}\right)\) | \(e\left(\frac{101}{196}\right)\) |
| \(\chi_{197}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{196}\right)\) | \(e\left(\frac{107}{196}\right)\) | \(e\left(\frac{19}{98}\right)\) | \(e\left(\frac{123}{196}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{15}{98}\right)\) | \(e\left(\frac{57}{196}\right)\) | \(e\left(\frac{9}{98}\right)\) | \(e\left(\frac{71}{98}\right)\) | \(e\left(\frac{159}{196}\right)\) |
| \(\chi_{197}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{169}{196}\right)\) | \(e\left(\frac{13}{196}\right)\) | \(e\left(\frac{71}{98}\right)\) | \(e\left(\frac{145}{196}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{87}{98}\right)\) | \(e\left(\frac{115}{196}\right)\) | \(e\left(\frac{13}{98}\right)\) | \(e\left(\frac{59}{98}\right)\) | \(e\left(\frac{1}{196}\right)\) |
| \(\chi_{197}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{196}\right)\) | \(e\left(\frac{109}{196}\right)\) | \(e\left(\frac{45}{98}\right)\) | \(e\left(\frac{85}{196}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{51}{98}\right)\) | \(e\left(\frac{135}{196}\right)\) | \(e\left(\frac{11}{98}\right)\) | \(e\left(\frac{65}{98}\right)\) | \(e\left(\frac{129}{196}\right)\) |
| \(\chi_{197}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{196}\right)\) | \(e\left(\frac{45}{196}\right)\) | \(e\left(\frac{95}{98}\right)\) | \(e\left(\frac{125}{196}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{75}{98}\right)\) | \(e\left(\frac{187}{196}\right)\) | \(e\left(\frac{45}{98}\right)\) | \(e\left(\frac{61}{98}\right)\) | \(e\left(\frac{109}{196}\right)\) |