from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4334, base_ring=CyclotomicField(196))
M = H._module
chi = DirichletCharacter(H, M([0,59]))
chi.galois_orbit()
[g,chi] = znchar(Mod(45,4334))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4334\) | |
| Conductor: | \(197\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(196\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 197.i | 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4334}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{196}\right)\) | \(e\left(\frac{155}{196}\right)\) | \(e\left(\frac{93}{98}\right)\) | \(e\left(\frac{95}{98}\right)\) | \(e\left(\frac{103}{196}\right)\) | \(e\left(\frac{27}{98}\right)\) | \(e\left(\frac{169}{196}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{85}{196}\right)\) | \(e\left(\frac{6}{49}\right)\) |
| \(\chi_{4334}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{196}\right)\) | \(e\left(\frac{141}{196}\right)\) | \(e\left(\frac{65}{98}\right)\) | \(e\left(\frac{39}{98}\right)\) | \(e\left(\frac{33}{196}\right)\) | \(e\left(\frac{41}{98}\right)\) | \(e\left(\frac{155}{196}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{71}{196}\right)\) | \(e\left(\frac{20}{49}\right)\) |
| \(\chi_{4334}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{155}{196}\right)\) | \(e\left(\frac{191}{196}\right)\) | \(e\left(\frac{95}{98}\right)\) | \(e\left(\frac{57}{98}\right)\) | \(e\left(\frac{3}{196}\right)\) | \(e\left(\frac{75}{98}\right)\) | \(e\left(\frac{121}{196}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{149}{196}\right)\) | \(e\left(\frac{33}{49}\right)\) |
| \(\chi_{4334}(111,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{196}\right)\) | \(e\left(\frac{73}{196}\right)\) | \(e\left(\frac{83}{98}\right)\) | \(e\left(\frac{89}{98}\right)\) | \(e\left(\frac{113}{196}\right)\) | \(e\left(\frac{81}{98}\right)\) | \(e\left(\frac{115}{196}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{59}{196}\right)\) | \(e\left(\frac{18}{49}\right)\) |
| \(\chi_{4334}(199,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{196}\right)\) | \(e\left(\frac{89}{196}\right)\) | \(e\left(\frac{73}{98}\right)\) | \(e\left(\frac{83}{98}\right)\) | \(e\left(\frac{25}{196}\right)\) | \(e\left(\frac{37}{98}\right)\) | \(e\left(\frac{159}{196}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{131}{196}\right)\) | \(e\left(\frac{30}{49}\right)\) |
| \(\chi_{4334}(243,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{196}\right)\) | \(e\left(\frac{185}{196}\right)\) | \(e\left(\frac{13}{98}\right)\) | \(e\left(\frac{47}{98}\right)\) | \(e\left(\frac{85}{196}\right)\) | \(e\left(\frac{67}{98}\right)\) | \(e\left(\frac{31}{196}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{171}{196}\right)\) | \(e\left(\frac{4}{49}\right)\) |
| \(\chi_{4334}(397,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{196}\right)\) | \(e\left(\frac{37}{196}\right)\) | \(e\left(\frac{81}{98}\right)\) | \(e\left(\frac{29}{98}\right)\) | \(e\left(\frac{17}{196}\right)\) | \(e\left(\frac{33}{98}\right)\) | \(e\left(\frac{163}{196}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{191}{196}\right)\) | \(e\left(\frac{40}{49}\right)\) |
| \(\chi_{4334}(485,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{196}\right)\) | \(e\left(\frac{127}{196}\right)\) | \(e\left(\frac{37}{98}\right)\) | \(e\left(\frac{81}{98}\right)\) | \(e\left(\frac{159}{196}\right)\) | \(e\left(\frac{55}{98}\right)\) | \(e\left(\frac{141}{196}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{57}{196}\right)\) | \(e\left(\frac{34}{49}\right)\) |
| \(\chi_{4334}(573,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{141}{196}\right)\) | \(e\left(\frac{65}{196}\right)\) | \(e\left(\frac{39}{98}\right)\) | \(e\left(\frac{43}{98}\right)\) | \(e\left(\frac{157}{196}\right)\) | \(e\left(\frac{5}{98}\right)\) | \(e\left(\frac{191}{196}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{23}{196}\right)\) | \(e\left(\frac{12}{49}\right)\) |
| \(\chi_{4334}(639,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{165}{196}\right)\) | \(e\left(\frac{1}{196}\right)\) | \(e\left(\frac{79}{98}\right)\) | \(e\left(\frac{67}{98}\right)\) | \(e\left(\frac{117}{196}\right)\) | \(e\left(\frac{83}{98}\right)\) | \(e\left(\frac{15}{196}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{127}{196}\right)\) | \(e\left(\frac{13}{49}\right)\) |
| \(\chi_{4334}(771,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{196}\right)\) | \(e\left(\frac{137}{196}\right)\) | \(e\left(\frac{43}{98}\right)\) | \(e\left(\frac{65}{98}\right)\) | \(e\left(\frac{153}{196}\right)\) | \(e\left(\frac{3}{98}\right)\) | \(e\left(\frac{95}{196}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{151}{196}\right)\) | \(e\left(\frac{17}{49}\right)\) |
| \(\chi_{4334}(793,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{196}\right)\) | \(e\left(\frac{81}{196}\right)\) | \(e\left(\frac{29}{98}\right)\) | \(e\left(\frac{37}{98}\right)\) | \(e\left(\frac{69}{196}\right)\) | \(e\left(\frac{59}{98}\right)\) | \(e\left(\frac{39}{196}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{95}{196}\right)\) | \(e\left(\frac{24}{49}\right)\) |
| \(\chi_{4334}(815,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{87}{196}\right)\) | \(e\left(\frac{111}{196}\right)\) | \(e\left(\frac{47}{98}\right)\) | \(e\left(\frac{87}{98}\right)\) | \(e\left(\frac{51}{196}\right)\) | \(e\left(\frac{1}{98}\right)\) | \(e\left(\frac{97}{196}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{181}{196}\right)\) | \(e\left(\frac{22}{49}\right)\) |
| \(\chi_{4334}(859,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{196}\right)\) | \(e\left(\frac{123}{196}\right)\) | \(e\left(\frac{15}{98}\right)\) | \(e\left(\frac{9}{98}\right)\) | \(e\left(\frac{83}{196}\right)\) | \(e\left(\frac{17}{98}\right)\) | \(e\left(\frac{81}{196}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{137}{196}\right)\) | \(e\left(\frac{31}{49}\right)\) |
| \(\chi_{4334}(903,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{196}\right)\) | \(e\left(\frac{177}{196}\right)\) | \(e\left(\frac{67}{98}\right)\) | \(e\left(\frac{1}{98}\right)\) | \(e\left(\frac{129}{196}\right)\) | \(e\left(\frac{89}{98}\right)\) | \(e\left(\frac{107}{196}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{135}{196}\right)\) | \(e\left(\frac{47}{49}\right)\) |
| \(\chi_{4334}(947,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{196}\right)\) | \(e\left(\frac{173}{196}\right)\) | \(e\left(\frac{45}{98}\right)\) | \(e\left(\frac{27}{98}\right)\) | \(e\left(\frac{53}{196}\right)\) | \(e\left(\frac{51}{98}\right)\) | \(e\left(\frac{47}{196}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{19}{196}\right)\) | \(e\left(\frac{44}{49}\right)\) |
| \(\chi_{4334}(1035,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{196}\right)\) | \(e\left(\frac{55}{196}\right)\) | \(e\left(\frac{33}{98}\right)\) | \(e\left(\frac{59}{98}\right)\) | \(e\left(\frac{163}{196}\right)\) | \(e\left(\frac{57}{98}\right)\) | \(e\left(\frac{41}{196}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{125}{196}\right)\) | \(e\left(\frac{29}{49}\right)\) |
| \(\chi_{4334}(1057,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{196}\right)\) | \(e\left(\frac{145}{196}\right)\) | \(e\left(\frac{87}{98}\right)\) | \(e\left(\frac{13}{98}\right)\) | \(e\left(\frac{109}{196}\right)\) | \(e\left(\frac{79}{98}\right)\) | \(e\left(\frac{19}{196}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{187}{196}\right)\) | \(e\left(\frac{23}{49}\right)\) |
| \(\chi_{4334}(1079,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{171}{196}\right)\) | \(e\left(\frac{83}{196}\right)\) | \(e\left(\frac{89}{98}\right)\) | \(e\left(\frac{73}{98}\right)\) | \(e\left(\frac{107}{196}\right)\) | \(e\left(\frac{29}{98}\right)\) | \(e\left(\frac{69}{196}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{153}{196}\right)\) | \(e\left(\frac{1}{49}\right)\) |
| \(\chi_{4334}(1255,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{196}\right)\) | \(e\left(\frac{85}{196}\right)\) | \(e\left(\frac{51}{98}\right)\) | \(e\left(\frac{11}{98}\right)\) | \(e\left(\frac{145}{196}\right)\) | \(e\left(\frac{97}{98}\right)\) | \(e\left(\frac{99}{196}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{15}{196}\right)\) | \(e\left(\frac{27}{49}\right)\) |
| \(\chi_{4334}(1277,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{196}\right)\) | \(e\left(\frac{67}{196}\right)\) | \(e\left(\frac{1}{98}\right)\) | \(e\left(\frac{79}{98}\right)\) | \(e\left(\frac{195}{196}\right)\) | \(e\left(\frac{73}{98}\right)\) | \(e\left(\frac{25}{196}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{81}{196}\right)\) | \(e\left(\frac{38}{49}\right)\) |
| \(\chi_{4334}(1299,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{75}{196}\right)\) | \(e\left(\frac{143}{196}\right)\) | \(e\left(\frac{27}{98}\right)\) | \(e\left(\frac{75}{98}\right)\) | \(e\left(\frac{71}{196}\right)\) | \(e\left(\frac{11}{98}\right)\) | \(e\left(\frac{185}{196}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{129}{196}\right)\) | \(e\left(\frac{46}{49}\right)\) |
| \(\chi_{4334}(1321,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{196}\right)\) | \(e\left(\frac{59}{196}\right)\) | \(e\left(\frac{55}{98}\right)\) | \(e\left(\frac{33}{98}\right)\) | \(e\left(\frac{43}{196}\right)\) | \(e\left(\frac{95}{98}\right)\) | \(e\left(\frac{101}{196}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{45}{196}\right)\) | \(e\left(\frac{32}{49}\right)\) |
| \(\chi_{4334}(1387,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{196}\right)\) | \(e\left(\frac{71}{196}\right)\) | \(e\left(\frac{23}{98}\right)\) | \(e\left(\frac{53}{98}\right)\) | \(e\left(\frac{75}{196}\right)\) | \(e\left(\frac{13}{98}\right)\) | \(e\left(\frac{85}{196}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{196}\right)\) | \(e\left(\frac{41}{49}\right)\) |
| \(\chi_{4334}(1409,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{196}\right)\) | \(e\left(\frac{11}{196}\right)\) | \(e\left(\frac{85}{98}\right)\) | \(e\left(\frac{51}{98}\right)\) | \(e\left(\frac{111}{196}\right)\) | \(e\left(\frac{31}{98}\right)\) | \(e\left(\frac{165}{196}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{25}{196}\right)\) | \(e\left(\frac{45}{49}\right)\) |
| \(\chi_{4334}(1431,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{183}{196}\right)\) | \(e\left(\frac{51}{196}\right)\) | \(e\left(\frac{11}{98}\right)\) | \(e\left(\frac{85}{98}\right)\) | \(e\left(\frac{87}{196}\right)\) | \(e\left(\frac{19}{98}\right)\) | \(e\left(\frac{177}{196}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{9}{196}\right)\) | \(e\left(\frac{26}{49}\right)\) |
| \(\chi_{4334}(1453,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{196}\right)\) | \(e\left(\frac{125}{196}\right)\) | \(e\left(\frac{75}{98}\right)\) | \(e\left(\frac{45}{98}\right)\) | \(e\left(\frac{121}{196}\right)\) | \(e\left(\frac{85}{98}\right)\) | \(e\left(\frac{111}{196}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{195}{196}\right)\) | \(e\left(\frac{8}{49}\right)\) |
| \(\chi_{4334}(1497,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{196}\right)\) | \(e\left(\frac{17}{196}\right)\) | \(e\left(\frac{69}{98}\right)\) | \(e\left(\frac{61}{98}\right)\) | \(e\left(\frac{29}{196}\right)\) | \(e\left(\frac{39}{98}\right)\) | \(e\left(\frac{59}{196}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{196}\right)\) | \(e\left(\frac{25}{49}\right)\) |
| \(\chi_{4334}(1519,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{169}{196}\right)\) | \(e\left(\frac{121}{196}\right)\) | \(e\left(\frac{53}{98}\right)\) | \(e\left(\frac{71}{98}\right)\) | \(e\left(\frac{45}{196}\right)\) | \(e\left(\frac{47}{98}\right)\) | \(e\left(\frac{51}{196}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{79}{196}\right)\) | \(e\left(\frac{5}{49}\right)\) |
| \(\chi_{4334}(1541,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{196}\right)\) | \(e\left(\frac{41}{196}\right)\) | \(e\left(\frac{5}{98}\right)\) | \(e\left(\frac{3}{98}\right)\) | \(e\left(\frac{93}{196}\right)\) | \(e\left(\frac{71}{98}\right)\) | \(e\left(\frac{27}{196}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{111}{196}\right)\) | \(e\left(\frac{43}{49}\right)\) |
| \(\chi_{4334}(1563,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{196}\right)\) | \(e\left(\frac{167}{196}\right)\) | \(e\left(\frac{61}{98}\right)\) | \(e\left(\frac{17}{98}\right)\) | \(e\left(\frac{135}{196}\right)\) | \(e\left(\frac{43}{98}\right)\) | \(e\left(\frac{153}{196}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{41}{196}\right)\) | \(e\left(\frac{15}{49}\right)\) |