from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4001, base_ring=CyclotomicField(160))
M = H._module
chi = DirichletCharacter(H, M([57]))
chi.galois_orbit()
[g,chi] = znchar(Mod(11,4001))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4001\) | |
| Conductor: | \(4001\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(160\) | 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: | $\Q(\zeta_{160})$ |
| Fixed field: | Number field defined by a degree 160 polynomial (not computed) |
First 31 of 64 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4001}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{57}{160}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{101}{160}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{21}{32}\right)\) |
| \(\chi_{4001}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{157}{160}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{121}{160}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{9}{32}\right)\) |
| \(\chi_{4001}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{39}{160}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{27}{160}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{32}\right)\) |
| \(\chi_{4001}(163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{109}{160}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{137}{160}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{25}{32}\right)\) |
| \(\chi_{4001}(233,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{1}{160}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{160}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{29}{32}\right)\) |
| \(\chi_{4001}(262,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{107}{160}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{111}{160}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{31}{32}\right)\) |
| \(\chi_{4001}(265,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{43}{160}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{79}{160}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{31}{32}\right)\) |
| \(\chi_{4001}(270,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{131}{160}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{103}{160}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{23}{32}\right)\) |
| \(\chi_{4001}(282,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{143}{160}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{99}{160}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{19}{32}\right)\) |
| \(\chi_{4001}(306,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{141}{160}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{73}{160}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{25}{32}\right)\) |
| \(\chi_{4001}(461,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{127}{160}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{51}{160}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{32}\right)\) |
| \(\chi_{4001}(521,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{147}{160}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{151}{160}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{32}\right)\) |
| \(\chi_{4001}(593,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{9}{160}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{117}{160}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{5}{32}\right)\) |
| \(\chi_{4001}(599,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{13}{160}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{160}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{25}{32}\right)\) |
| \(\chi_{4001}(770,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{37}{160}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{160}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{32}\right)\) |
| \(\chi_{4001}(974,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{129}{160}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{77}{160}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{29}{32}\right)\) |
| \(\chi_{4001}(975,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{21}{160}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{113}{160}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{32}\right)\) |
| \(\chi_{4001}(1030,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{59}{160}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{127}{160}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{15}{32}\right)\) |
| \(\chi_{4001}(1105,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{31}{160}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{83}{160}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{32}\right)\) |
| \(\chi_{4001}(1147,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{151}{160}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{43}{160}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{27}{32}\right)\) |
| \(\chi_{4001}(1331,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{11}{160}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{143}{160}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{31}{32}\right)\) |
| \(\chi_{4001}(1415,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{121}{160}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{133}{160}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{21}{32}\right)\) |
| \(\chi_{4001}(1455,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{103}{160}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{59}{160}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{11}{32}\right)\) |
| \(\chi_{4001}(1500,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{149}{160}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{17}{160}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{32}\right)\) |
| \(\chi_{4001}(1634,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{53}{160}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{49}{160}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{32}\right)\) |
| \(\chi_{4001}(1649,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{113}{160}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{29}{160}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{13}{32}\right)\) |
| \(\chi_{4001}(1665,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{7}{160}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{91}{160}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{32}\right)\) |
| \(\chi_{4001}(1700,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{159}{160}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{147}{160}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{32}\right)\) |
| \(\chi_{4001}(1739,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{19}{160}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{87}{160}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{32}\right)\) |
| \(\chi_{4001}(1825,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{83}{160}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{119}{160}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{32}\right)\) |
| \(\chi_{4001}(1887,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{17}{160}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{61}{160}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{13}{32}\right)\) |