from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1681, base_ring=CyclotomicField(410))
M = H._module
chi = DirichletCharacter(H, M([232]))
chi.galois_orbit()
[g,chi] = znchar(Mod(10,1681))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1681\) | |
| Conductor: | \(1681\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(205\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{205})$ |
| Fixed field: | Number field defined by a degree 205 polynomial (not computed) |
First 31 of 160 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1681}(10,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{205}\right)\) | \(e\left(\frac{23}{41}\right)\) | \(e\left(\frac{2}{205}\right)\) | \(e\left(\frac{22}{205}\right)\) | \(e\left(\frac{116}{205}\right)\) | \(e\left(\frac{64}{205}\right)\) | \(e\left(\frac{3}{205}\right)\) | \(e\left(\frac{5}{41}\right)\) | \(e\left(\frac{23}{205}\right)\) | \(e\left(\frac{173}{205}\right)\) |
| \(\chi_{1681}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{205}\right)\) | \(e\left(\frac{4}{41}\right)\) | \(e\left(\frac{36}{205}\right)\) | \(e\left(\frac{191}{205}\right)\) | \(e\left(\frac{38}{205}\right)\) | \(e\left(\frac{127}{205}\right)\) | \(e\left(\frac{54}{205}\right)\) | \(e\left(\frac{8}{41}\right)\) | \(e\left(\frac{4}{205}\right)\) | \(e\left(\frac{39}{205}\right)\) |
| \(\chi_{1681}(18,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{117}{205}\right)\) | \(e\left(\frac{26}{41}\right)\) | \(e\left(\frac{29}{205}\right)\) | \(e\left(\frac{114}{205}\right)\) | \(e\left(\frac{42}{205}\right)\) | \(e\left(\frac{108}{205}\right)\) | \(e\left(\frac{146}{205}\right)\) | \(e\left(\frac{11}{41}\right)\) | \(e\left(\frac{26}{205}\right)\) | \(e\left(\frac{151}{205}\right)\) |
| \(\chi_{1681}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{114}{205}\right)\) | \(e\left(\frac{39}{41}\right)\) | \(e\left(\frac{23}{205}\right)\) | \(e\left(\frac{48}{205}\right)\) | \(e\left(\frac{104}{205}\right)\) | \(e\left(\frac{121}{205}\right)\) | \(e\left(\frac{137}{205}\right)\) | \(e\left(\frac{37}{41}\right)\) | \(e\left(\frac{162}{205}\right)\) | \(e\left(\frac{42}{205}\right)\) |
| \(\chi_{1681}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{205}\right)\) | \(e\left(\frac{5}{41}\right)\) | \(e\left(\frac{86}{205}\right)\) | \(e\left(\frac{126}{205}\right)\) | \(e\left(\frac{68}{205}\right)\) | \(e\left(\frac{87}{205}\right)\) | \(e\left(\frac{129}{205}\right)\) | \(e\left(\frac{10}{41}\right)\) | \(e\left(\frac{169}{205}\right)\) | \(e\left(\frac{59}{205}\right)\) |
| \(\chi_{1681}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{162}{205}\right)\) | \(e\left(\frac{36}{41}\right)\) | \(e\left(\frac{119}{205}\right)\) | \(e\left(\frac{79}{205}\right)\) | \(e\left(\frac{137}{205}\right)\) | \(e\left(\frac{118}{205}\right)\) | \(e\left(\frac{76}{205}\right)\) | \(e\left(\frac{31}{41}\right)\) | \(e\left(\frac{36}{205}\right)\) | \(e\left(\frac{146}{205}\right)\) |
| \(\chi_{1681}(78,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{205}\right)\) | \(e\left(\frac{35}{41}\right)\) | \(e\left(\frac{28}{205}\right)\) | \(e\left(\frac{103}{205}\right)\) | \(e\left(\frac{189}{205}\right)\) | \(e\left(\frac{76}{205}\right)\) | \(e\left(\frac{42}{205}\right)\) | \(e\left(\frac{29}{41}\right)\) | \(e\left(\frac{117}{205}\right)\) | \(e\left(\frac{167}{205}\right)\) |
| \(\chi_{1681}(92,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{205}\right)\) | \(e\left(\frac{18}{41}\right)\) | \(e\left(\frac{162}{205}\right)\) | \(e\left(\frac{142}{205}\right)\) | \(e\left(\frac{171}{205}\right)\) | \(e\left(\frac{59}{205}\right)\) | \(e\left(\frac{38}{205}\right)\) | \(e\left(\frac{36}{41}\right)\) | \(e\left(\frac{18}{205}\right)\) | \(e\left(\frac{73}{205}\right)\) |
| \(\chi_{1681}(98,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{68}{205}\right)\) | \(e\left(\frac{6}{41}\right)\) | \(e\left(\frac{136}{205}\right)\) | \(e\left(\frac{61}{205}\right)\) | \(e\left(\frac{98}{205}\right)\) | \(e\left(\frac{47}{205}\right)\) | \(e\left(\frac{204}{205}\right)\) | \(e\left(\frac{12}{41}\right)\) | \(e\left(\frac{129}{205}\right)\) | \(e\left(\frac{79}{205}\right)\) |
| \(\chi_{1681}(100,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{205}\right)\) | \(e\left(\frac{5}{41}\right)\) | \(e\left(\frac{4}{205}\right)\) | \(e\left(\frac{44}{205}\right)\) | \(e\left(\frac{27}{205}\right)\) | \(e\left(\frac{128}{205}\right)\) | \(e\left(\frac{6}{205}\right)\) | \(e\left(\frac{10}{41}\right)\) | \(e\left(\frac{46}{205}\right)\) | \(e\left(\frac{141}{205}\right)\) |
| \(\chi_{1681}(119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{205}\right)\) | \(e\left(\frac{31}{41}\right)\) | \(e\left(\frac{33}{205}\right)\) | \(e\left(\frac{158}{205}\right)\) | \(e\left(\frac{69}{205}\right)\) | \(e\left(\frac{31}{205}\right)\) | \(e\left(\frac{152}{205}\right)\) | \(e\left(\frac{21}{41}\right)\) | \(e\left(\frac{72}{205}\right)\) | \(e\left(\frac{87}{205}\right)\) |
| \(\chi_{1681}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{205}\right)\) | \(e\left(\frac{36}{41}\right)\) | \(e\left(\frac{37}{205}\right)\) | \(e\left(\frac{202}{205}\right)\) | \(e\left(\frac{96}{205}\right)\) | \(e\left(\frac{159}{205}\right)\) | \(e\left(\frac{158}{205}\right)\) | \(e\left(\frac{31}{41}\right)\) | \(e\left(\frac{118}{205}\right)\) | \(e\left(\frac{23}{205}\right)\) |
| \(\chi_{1681}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{93}{205}\right)\) | \(e\left(\frac{7}{41}\right)\) | \(e\left(\frac{186}{205}\right)\) | \(e\left(\frac{201}{205}\right)\) | \(e\left(\frac{128}{205}\right)\) | \(e\left(\frac{7}{205}\right)\) | \(e\left(\frac{74}{205}\right)\) | \(e\left(\frac{14}{41}\right)\) | \(e\left(\frac{89}{205}\right)\) | \(e\left(\frac{99}{205}\right)\) |
| \(\chi_{1681}(141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{205}\right)\) | \(e\left(\frac{15}{41}\right)\) | \(e\left(\frac{94}{205}\right)\) | \(e\left(\frac{9}{205}\right)\) | \(e\left(\frac{122}{205}\right)\) | \(e\left(\frac{138}{205}\right)\) | \(e\left(\frac{141}{205}\right)\) | \(e\left(\frac{30}{41}\right)\) | \(e\left(\frac{56}{205}\right)\) | \(e\left(\frac{136}{205}\right)\) |
| \(\chi_{1681}(160,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{205}\right)\) | \(e\left(\frac{27}{41}\right)\) | \(e\left(\frac{38}{205}\right)\) | \(e\left(\frac{8}{205}\right)\) | \(e\left(\frac{154}{205}\right)\) | \(e\left(\frac{191}{205}\right)\) | \(e\left(\frac{57}{205}\right)\) | \(e\left(\frac{13}{41}\right)\) | \(e\left(\frac{27}{205}\right)\) | \(e\left(\frac{7}{205}\right)\) |
| \(\chi_{1681}(174,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{205}\right)\) | \(e\left(\frac{13}{41}\right)\) | \(e\left(\frac{117}{205}\right)\) | \(e\left(\frac{57}{205}\right)\) | \(e\left(\frac{21}{205}\right)\) | \(e\left(\frac{54}{205}\right)\) | \(e\left(\frac{73}{205}\right)\) | \(e\left(\frac{26}{41}\right)\) | \(e\left(\frac{13}{205}\right)\) | \(e\left(\frac{178}{205}\right)\) |
| \(\chi_{1681}(180,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{118}{205}\right)\) | \(e\left(\frac{8}{41}\right)\) | \(e\left(\frac{31}{205}\right)\) | \(e\left(\frac{136}{205}\right)\) | \(e\left(\frac{158}{205}\right)\) | \(e\left(\frac{172}{205}\right)\) | \(e\left(\frac{149}{205}\right)\) | \(e\left(\frac{16}{41}\right)\) | \(e\left(\frac{49}{205}\right)\) | \(e\left(\frac{119}{205}\right)\) |
| \(\chi_{1681}(182,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{92}{205}\right)\) | \(e\left(\frac{25}{41}\right)\) | \(e\left(\frac{184}{205}\right)\) | \(e\left(\frac{179}{205}\right)\) | \(e\left(\frac{12}{205}\right)\) | \(e\left(\frac{148}{205}\right)\) | \(e\left(\frac{71}{205}\right)\) | \(e\left(\frac{9}{41}\right)\) | \(e\left(\frac{66}{205}\right)\) | \(e\left(\frac{131}{205}\right)\) |
| \(\chi_{1681}(201,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{124}{205}\right)\) | \(e\left(\frac{23}{41}\right)\) | \(e\left(\frac{43}{205}\right)\) | \(e\left(\frac{63}{205}\right)\) | \(e\left(\frac{34}{205}\right)\) | \(e\left(\frac{146}{205}\right)\) | \(e\left(\frac{167}{205}\right)\) | \(e\left(\frac{5}{41}\right)\) | \(e\left(\frac{187}{205}\right)\) | \(e\left(\frac{132}{205}\right)\) |
| \(\chi_{1681}(215,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{201}{205}\right)\) | \(e\left(\frac{31}{41}\right)\) | \(e\left(\frac{197}{205}\right)\) | \(e\left(\frac{117}{205}\right)\) | \(e\left(\frac{151}{205}\right)\) | \(e\left(\frac{154}{205}\right)\) | \(e\left(\frac{193}{205}\right)\) | \(e\left(\frac{21}{41}\right)\) | \(e\left(\frac{113}{205}\right)\) | \(e\left(\frac{128}{205}\right)\) |
| \(\chi_{1681}(221,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{205}\right)\) | \(e\left(\frac{9}{41}\right)\) | \(e\left(\frac{81}{205}\right)\) | \(e\left(\frac{71}{205}\right)\) | \(e\left(\frac{188}{205}\right)\) | \(e\left(\frac{132}{205}\right)\) | \(e\left(\frac{19}{205}\right)\) | \(e\left(\frac{18}{41}\right)\) | \(e\left(\frac{9}{205}\right)\) | \(e\left(\frac{139}{205}\right)\) |
| \(\chi_{1681}(223,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{205}\right)\) | \(e\left(\frac{35}{41}\right)\) | \(e\left(\frac{69}{205}\right)\) | \(e\left(\frac{144}{205}\right)\) | \(e\left(\frac{107}{205}\right)\) | \(e\left(\frac{158}{205}\right)\) | \(e\left(\frac{1}{205}\right)\) | \(e\left(\frac{29}{41}\right)\) | \(e\left(\frac{76}{205}\right)\) | \(e\left(\frac{126}{205}\right)\) |
| \(\chi_{1681}(242,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{205}\right)\) | \(e\left(\frac{19}{41}\right)\) | \(e\left(\frac{48}{205}\right)\) | \(e\left(\frac{118}{205}\right)\) | \(e\left(\frac{119}{205}\right)\) | \(e\left(\frac{101}{205}\right)\) | \(e\left(\frac{72}{205}\right)\) | \(e\left(\frac{38}{41}\right)\) | \(e\left(\frac{142}{205}\right)\) | \(e\left(\frac{52}{205}\right)\) |
| \(\chi_{1681}(256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{36}{205}\right)\) | \(e\left(\frac{8}{41}\right)\) | \(e\left(\frac{72}{205}\right)\) | \(e\left(\frac{177}{205}\right)\) | \(e\left(\frac{76}{205}\right)\) | \(e\left(\frac{49}{205}\right)\) | \(e\left(\frac{108}{205}\right)\) | \(e\left(\frac{16}{41}\right)\) | \(e\left(\frac{8}{205}\right)\) | \(e\left(\frac{78}{205}\right)\) |
| \(\chi_{1681}(262,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{168}{205}\right)\) | \(e\left(\frac{10}{41}\right)\) | \(e\left(\frac{131}{205}\right)\) | \(e\left(\frac{6}{205}\right)\) | \(e\left(\frac{13}{205}\right)\) | \(e\left(\frac{92}{205}\right)\) | \(e\left(\frac{94}{205}\right)\) | \(e\left(\frac{20}{41}\right)\) | \(e\left(\frac{174}{205}\right)\) | \(e\left(\frac{159}{205}\right)\) |
| \(\chi_{1681}(264,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{182}{205}\right)\) | \(e\left(\frac{4}{41}\right)\) | \(e\left(\frac{159}{205}\right)\) | \(e\left(\frac{109}{205}\right)\) | \(e\left(\frac{202}{205}\right)\) | \(e\left(\frac{168}{205}\right)\) | \(e\left(\frac{136}{205}\right)\) | \(e\left(\frac{8}{41}\right)\) | \(e\left(\frac{86}{205}\right)\) | \(e\left(\frac{121}{205}\right)\) |
| \(\chi_{1681}(283,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{129}{205}\right)\) | \(e\left(\frac{15}{41}\right)\) | \(e\left(\frac{53}{205}\right)\) | \(e\left(\frac{173}{205}\right)\) | \(e\left(\frac{204}{205}\right)\) | \(e\left(\frac{56}{205}\right)\) | \(e\left(\frac{182}{205}\right)\) | \(e\left(\frac{30}{41}\right)\) | \(e\left(\frac{97}{205}\right)\) | \(e\left(\frac{177}{205}\right)\) |
| \(\chi_{1681}(297,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{76}{205}\right)\) | \(e\left(\frac{26}{41}\right)\) | \(e\left(\frac{152}{205}\right)\) | \(e\left(\frac{32}{205}\right)\) | \(e\left(\frac{1}{205}\right)\) | \(e\left(\frac{149}{205}\right)\) | \(e\left(\frac{23}{205}\right)\) | \(e\left(\frac{11}{41}\right)\) | \(e\left(\frac{108}{205}\right)\) | \(e\left(\frac{28}{205}\right)\) |
| \(\chi_{1681}(303,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{205}\right)\) | \(e\left(\frac{11}{41}\right)\) | \(e\left(\frac{181}{205}\right)\) | \(e\left(\frac{146}{205}\right)\) | \(e\left(\frac{43}{205}\right)\) | \(e\left(\frac{52}{205}\right)\) | \(e\left(\frac{169}{205}\right)\) | \(e\left(\frac{22}{41}\right)\) | \(e\left(\frac{134}{205}\right)\) | \(e\left(\frac{179}{205}\right)\) |
| \(\chi_{1681}(305,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{205}\right)\) | \(e\left(\frac{14}{41}\right)\) | \(e\left(\frac{44}{205}\right)\) | \(e\left(\frac{74}{205}\right)\) | \(e\left(\frac{92}{205}\right)\) | \(e\left(\frac{178}{205}\right)\) | \(e\left(\frac{66}{205}\right)\) | \(e\left(\frac{28}{41}\right)\) | \(e\left(\frac{96}{205}\right)\) | \(e\left(\frac{116}{205}\right)\) |
| \(\chi_{1681}(324,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{205}\right)\) | \(e\left(\frac{11}{41}\right)\) | \(e\left(\frac{58}{205}\right)\) | \(e\left(\frac{23}{205}\right)\) | \(e\left(\frac{84}{205}\right)\) | \(e\left(\frac{11}{205}\right)\) | \(e\left(\frac{87}{205}\right)\) | \(e\left(\frac{22}{41}\right)\) | \(e\left(\frac{52}{205}\right)\) | \(e\left(\frac{97}{205}\right)\) |