from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3997, base_ring=CyclotomicField(570))
M = H._module
chi = DirichletCharacter(H, M([475,422]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,3997))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3997\) | |
| Conductor: | \(3997\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(570\) | 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: | $\Q(\zeta_{285})$ |
| Fixed field: | Number field defined by a degree 570 polynomial (not computed) |
First 31 of 144 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3997}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{109}{190}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{113}{190}\right)\) | \(e\left(\frac{139}{190}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{14}{95}\right)\) | \(e\left(\frac{143}{190}\right)\) | \(e\left(\frac{71}{95}\right)\) | \(e\left(\frac{169}{190}\right)\) |
| \(\chi_{3997}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{7}{190}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{9}{190}\right)\) | \(e\left(\frac{117}{190}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{7}{95}\right)\) | \(e\left(\frac{119}{190}\right)\) | \(e\left(\frac{83}{95}\right)\) | \(e\left(\frac{37}{190}\right)\) |
| \(\chi_{3997}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{173}{190}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{141}{190}\right)\) | \(e\left(\frac{123}{190}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{78}{95}\right)\) | \(e\left(\frac{91}{190}\right)\) | \(e\left(\frac{2}{95}\right)\) | \(e\left(\frac{73}{190}\right)\) |
| \(\chi_{3997}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{151}{190}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{167}{190}\right)\) | \(e\left(\frac{81}{190}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{56}{95}\right)\) | \(e\left(\frac{97}{190}\right)\) | \(e\left(\frac{94}{95}\right)\) | \(e\left(\frac{11}{190}\right)\) |
| \(\chi_{3997}(66,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{113}{190}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{91}{190}\right)\) | \(e\left(\frac{43}{190}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{18}{95}\right)\) | \(e\left(\frac{21}{190}\right)\) | \(e\left(\frac{37}{95}\right)\) | \(e\left(\frac{163}{190}\right)\) |
| \(\chi_{3997}(80,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{139}{190}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{43}{190}\right)\) | \(e\left(\frac{179}{190}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{44}{95}\right)\) | \(e\left(\frac{83}{190}\right)\) | \(e\left(\frac{6}{95}\right)\) | \(e\left(\frac{29}{190}\right)\) |
| \(\chi_{3997}(96,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{117}{190}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{69}{190}\right)\) | \(e\left(\frac{137}{190}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{22}{95}\right)\) | \(e\left(\frac{89}{190}\right)\) | \(e\left(\frac{3}{95}\right)\) | \(e\left(\frac{157}{190}\right)\) |
| \(\chi_{3997}(157,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{11}{190}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{177}{190}\right)\) | \(e\left(\frac{21}{190}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{11}{95}\right)\) | \(e\left(\frac{187}{190}\right)\) | \(e\left(\frac{49}{95}\right)\) | \(e\left(\frac{31}{190}\right)\) |
| \(\chi_{3997}(159,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{53}{190}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{41}{190}\right)\) | \(e\left(\frac{153}{190}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{53}{95}\right)\) | \(e\left(\frac{141}{190}\right)\) | \(e\left(\frac{72}{95}\right)\) | \(e\left(\frac{63}{190}\right)\) |
| \(\chi_{3997}(178,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{47}{190}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{169}{190}\right)\) | \(e\left(\frac{107}{190}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{47}{95}\right)\) | \(e\left(\frac{39}{190}\right)\) | \(e\left(\frac{28}{95}\right)\) | \(e\left(\frac{167}{190}\right)\) |
| \(\chi_{3997}(180,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{93}{190}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{11}{190}\right)\) | \(e\left(\frac{143}{190}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{93}{95}\right)\) | \(e\left(\frac{61}{190}\right)\) | \(e\left(\frac{17}{95}\right)\) | \(e\left(\frac{3}{190}\right)\) |
| \(\chi_{3997}(185,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{93}{190}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{11}{190}\right)\) | \(e\left(\frac{143}{190}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{93}{95}\right)\) | \(e\left(\frac{61}{190}\right)\) | \(e\left(\frac{17}{95}\right)\) | \(e\left(\frac{3}{190}\right)\) |
| \(\chi_{3997}(222,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{71}{190}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{37}{190}\right)\) | \(e\left(\frac{101}{190}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{71}{95}\right)\) | \(e\left(\frac{67}{190}\right)\) | \(e\left(\frac{14}{95}\right)\) | \(e\left(\frac{131}{190}\right)\) |
| \(\chi_{3997}(227,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{63}{190}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{81}{190}\right)\) | \(e\left(\frac{103}{190}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{63}{95}\right)\) | \(e\left(\frac{121}{190}\right)\) | \(e\left(\frac{82}{95}\right)\) | \(e\left(\frac{143}{190}\right)\) |
| \(\chi_{3997}(264,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{33}{190}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{151}{190}\right)\) | \(e\left(\frac{63}{190}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{33}{95}\right)\) | \(e\left(\frac{181}{190}\right)\) | \(e\left(\frac{52}{95}\right)\) | \(e\left(\frac{93}{190}\right)\) |
| \(\chi_{3997}(320,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{59}{190}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{103}{190}\right)\) | \(e\left(\frac{9}{190}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{59}{95}\right)\) | \(e\left(\frac{53}{190}\right)\) | \(e\left(\frac{21}{95}\right)\) | \(e\left(\frac{149}{190}\right)\) |
| \(\chi_{3997}(458,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{73}{190}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{121}{190}\right)\) | \(e\left(\frac{53}{190}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{73}{95}\right)\) | \(e\left(\frac{101}{190}\right)\) | \(e\left(\frac{92}{95}\right)\) | \(e\left(\frac{33}{190}\right)\) |
| \(\chi_{3997}(495,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{9}{190}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{93}{190}\right)\) | \(e\left(\frac{69}{190}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{9}{95}\right)\) | \(e\left(\frac{153}{190}\right)\) | \(e\left(\frac{66}{95}\right)\) | \(e\left(\frac{129}{190}\right)\) |
| \(\chi_{3997}(584,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{149}{190}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{83}{190}\right)\) | \(e\left(\frac{129}{190}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{54}{95}\right)\) | \(e\left(\frac{63}{190}\right)\) | \(e\left(\frac{16}{95}\right)\) | \(e\left(\frac{109}{190}\right)\) |
| \(\chi_{3997}(605,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{161}{190}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{17}{190}\right)\) | \(e\left(\frac{31}{190}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{66}{95}\right)\) | \(e\left(\frac{77}{190}\right)\) | \(e\left(\frac{9}{95}\right)\) | \(e\left(\frac{91}{190}\right)\) |
| \(\chi_{3997}(628,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{121}{190}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{47}{190}\right)\) | \(e\left(\frac{41}{190}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{26}{95}\right)\) | \(e\left(\frac{157}{190}\right)\) | \(e\left(\frac{64}{95}\right)\) | \(e\left(\frac{151}{190}\right)\) |
| \(\chi_{3997}(654,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{87}{190}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{139}{190}\right)\) | \(e\left(\frac{97}{190}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{87}{95}\right)\) | \(e\left(\frac{149}{190}\right)\) | \(e\left(\frac{68}{95}\right)\) | \(e\left(\frac{107}{190}\right)\) |
| \(\chi_{3997}(668,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{129}{190}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{3}{190}\right)\) | \(e\left(\frac{39}{190}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{34}{95}\right)\) | \(e\left(\frac{103}{190}\right)\) | \(e\left(\frac{91}{95}\right)\) | \(e\left(\frac{139}{190}\right)\) |
| \(\chi_{3997}(712,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{157}{190}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{39}{190}\right)\) | \(e\left(\frac{127}{190}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{62}{95}\right)\) | \(e\left(\frac{9}{190}\right)\) | \(e\left(\frac{43}{95}\right)\) | \(e\left(\frac{97}{190}\right)\) |
| \(\chi_{3997}(726,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{139}{190}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{43}{190}\right)\) | \(e\left(\frac{179}{190}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{44}{95}\right)\) | \(e\left(\frac{83}{190}\right)\) | \(e\left(\frac{6}{95}\right)\) | \(e\left(\frac{29}{190}\right)\) |
| \(\chi_{3997}(740,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{13}{190}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{71}{190}\right)\) | \(e\left(\frac{163}{190}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{13}{95}\right)\) | \(e\left(\frac{31}{190}\right)\) | \(e\left(\frac{32}{95}\right)\) | \(e\left(\frac{123}{190}\right)\) |
| \(\chi_{3997}(761,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{63}{190}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{81}{190}\right)\) | \(e\left(\frac{103}{190}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{63}{95}\right)\) | \(e\left(\frac{121}{190}\right)\) | \(e\left(\frac{82}{95}\right)\) | \(e\left(\frac{143}{190}\right)\) |
| \(\chi_{3997}(773,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{59}{190}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{103}{190}\right)\) | \(e\left(\frac{9}{190}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{59}{95}\right)\) | \(e\left(\frac{53}{190}\right)\) | \(e\left(\frac{21}{95}\right)\) | \(e\left(\frac{149}{190}\right)\) |
| \(\chi_{3997}(794,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{177}{190}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{119}{190}\right)\) | \(e\left(\frac{27}{190}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{82}{95}\right)\) | \(e\left(\frac{159}{190}\right)\) | \(e\left(\frac{63}{95}\right)\) | \(e\left(\frac{67}{190}\right)\) |
| \(\chi_{3997}(864,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{181}{190}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{97}{190}\right)\) | \(e\left(\frac{121}{190}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{86}{95}\right)\) | \(e\left(\frac{37}{190}\right)\) | \(e\left(\frac{29}{95}\right)\) | \(e\left(\frac{61}{190}\right)\) |
| \(\chi_{3997}(866,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{129}{190}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{3}{190}\right)\) | \(e\left(\frac{39}{190}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{34}{95}\right)\) | \(e\left(\frac{103}{190}\right)\) | \(e\left(\frac{91}{95}\right)\) | \(e\left(\frac{139}{190}\right)\) |