from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6724, base_ring=CyclotomicField(164))
M = H._module
chi = DirichletCharacter(H, M([0,67]))
chi.galois_orbit()
[g,chi] = znchar(Mod(9,6724))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6724\) | |
| Conductor: | \(1681\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(164\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1681.k | 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: | $\Q(\zeta_{164})$ |
| Fixed field: | Number field defined by a degree 164 polynomial (not computed) |
First 31 of 80 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6724}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{141}{164}\right)\) | \(e\left(\frac{47}{82}\right)\) | \(e\left(\frac{61}{164}\right)\) | \(e\left(\frac{59}{82}\right)\) | \(e\left(\frac{113}{164}\right)\) | \(e\left(\frac{49}{164}\right)\) | \(e\left(\frac{71}{164}\right)\) | \(e\left(\frac{23}{164}\right)\) | \(e\left(\frac{63}{164}\right)\) | \(e\left(\frac{19}{82}\right)\) |
| \(\chi_{6724}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{164}\right)\) | \(e\left(\frac{21}{82}\right)\) | \(e\left(\frac{111}{164}\right)\) | \(e\left(\frac{63}{82}\right)\) | \(e\left(\frac{47}{164}\right)\) | \(e\left(\frac{151}{164}\right)\) | \(e\left(\frac{105}{164}\right)\) | \(e\left(\frac{101}{164}\right)\) | \(e\left(\frac{77}{164}\right)\) | \(e\left(\frac{5}{82}\right)\) |
| \(\chi_{6724}(173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{164}\right)\) | \(e\left(\frac{17}{82}\right)\) | \(e\left(\frac{125}{164}\right)\) | \(e\left(\frac{51}{82}\right)\) | \(e\left(\frac{81}{164}\right)\) | \(e\left(\frac{9}{164}\right)\) | \(e\left(\frac{3}{164}\right)\) | \(e\left(\frac{31}{164}\right)\) | \(e\left(\frac{35}{164}\right)\) | \(e\left(\frac{47}{82}\right)\) |
| \(\chi_{6724}(237,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{164}\right)\) | \(e\left(\frac{51}{82}\right)\) | \(e\left(\frac{47}{164}\right)\) | \(e\left(\frac{71}{82}\right)\) | \(e\left(\frac{79}{164}\right)\) | \(e\left(\frac{27}{164}\right)\) | \(e\left(\frac{9}{164}\right)\) | \(e\left(\frac{93}{164}\right)\) | \(e\left(\frac{105}{164}\right)\) | \(e\left(\frac{59}{82}\right)\) |
| \(\chi_{6724}(337,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{164}\right)\) | \(e\left(\frac{69}{82}\right)\) | \(e\left(\frac{25}{164}\right)\) | \(e\left(\frac{43}{82}\right)\) | \(e\left(\frac{49}{164}\right)\) | \(e\left(\frac{133}{164}\right)\) | \(e\left(\frac{99}{164}\right)\) | \(e\left(\frac{39}{164}\right)\) | \(e\left(\frac{7}{164}\right)\) | \(e\left(\frac{75}{82}\right)\) |
| \(\chi_{6724}(401,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{164}\right)\) | \(e\left(\frac{81}{82}\right)\) | \(e\left(\frac{147}{164}\right)\) | \(e\left(\frac{79}{82}\right)\) | \(e\left(\frac{111}{164}\right)\) | \(e\left(\frac{67}{164}\right)\) | \(e\left(\frac{77}{164}\right)\) | \(e\left(\frac{85}{164}\right)\) | \(e\left(\frac{133}{164}\right)\) | \(e\left(\frac{31}{82}\right)\) |
| \(\chi_{6724}(501,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{117}{164}\right)\) | \(e\left(\frac{39}{82}\right)\) | \(e\left(\frac{89}{164}\right)\) | \(e\left(\frac{35}{82}\right)\) | \(e\left(\frac{17}{164}\right)\) | \(e\left(\frac{93}{164}\right)\) | \(e\left(\frac{31}{164}\right)\) | \(e\left(\frac{47}{164}\right)\) | \(e\left(\frac{143}{164}\right)\) | \(e\left(\frac{21}{82}\right)\) |
| \(\chi_{6724}(565,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{87}{164}\right)\) | \(e\left(\frac{29}{82}\right)\) | \(e\left(\frac{83}{164}\right)\) | \(e\left(\frac{5}{82}\right)\) | \(e\left(\frac{143}{164}\right)\) | \(e\left(\frac{107}{164}\right)\) | \(e\left(\frac{145}{164}\right)\) | \(e\left(\frac{77}{164}\right)\) | \(e\left(\frac{161}{164}\right)\) | \(e\left(\frac{3}{82}\right)\) |
| \(\chi_{6724}(665,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{164}\right)\) | \(e\left(\frac{9}{82}\right)\) | \(e\left(\frac{153}{164}\right)\) | \(e\left(\frac{27}{82}\right)\) | \(e\left(\frac{149}{164}\right)\) | \(e\left(\frac{53}{164}\right)\) | \(e\left(\frac{127}{164}\right)\) | \(e\left(\frac{55}{164}\right)\) | \(e\left(\frac{115}{164}\right)\) | \(e\left(\frac{49}{82}\right)\) |
| \(\chi_{6724}(729,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{164}\right)\) | \(e\left(\frac{59}{82}\right)\) | \(e\left(\frac{19}{164}\right)\) | \(e\left(\frac{13}{82}\right)\) | \(e\left(\frac{11}{164}\right)\) | \(e\left(\frac{147}{164}\right)\) | \(e\left(\frac{49}{164}\right)\) | \(e\left(\frac{69}{164}\right)\) | \(e\left(\frac{25}{164}\right)\) | \(e\left(\frac{57}{82}\right)\) |
| \(\chi_{6724}(829,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{164}\right)\) | \(e\left(\frac{61}{82}\right)\) | \(e\left(\frac{53}{164}\right)\) | \(e\left(\frac{19}{82}\right)\) | \(e\left(\frac{117}{164}\right)\) | \(e\left(\frac{13}{164}\right)\) | \(e\left(\frac{59}{164}\right)\) | \(e\left(\frac{63}{164}\right)\) | \(e\left(\frac{87}{164}\right)\) | \(e\left(\frac{77}{82}\right)\) |
| \(\chi_{6724}(893,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{164}\right)\) | \(e\left(\frac{7}{82}\right)\) | \(e\left(\frac{119}{164}\right)\) | \(e\left(\frac{21}{82}\right)\) | \(e\left(\frac{43}{164}\right)\) | \(e\left(\frac{23}{164}\right)\) | \(e\left(\frac{117}{164}\right)\) | \(e\left(\frac{61}{164}\right)\) | \(e\left(\frac{53}{164}\right)\) | \(e\left(\frac{29}{82}\right)\) |
| \(\chi_{6724}(993,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{93}{164}\right)\) | \(e\left(\frac{31}{82}\right)\) | \(e\left(\frac{117}{164}\right)\) | \(e\left(\frac{11}{82}\right)\) | \(e\left(\frac{85}{164}\right)\) | \(e\left(\frac{137}{164}\right)\) | \(e\left(\frac{155}{164}\right)\) | \(e\left(\frac{71}{164}\right)\) | \(e\left(\frac{59}{164}\right)\) | \(e\left(\frac{23}{82}\right)\) |
| \(\chi_{6724}(1057,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{111}{164}\right)\) | \(e\left(\frac{37}{82}\right)\) | \(e\left(\frac{55}{164}\right)\) | \(e\left(\frac{29}{82}\right)\) | \(e\left(\frac{75}{164}\right)\) | \(e\left(\frac{63}{164}\right)\) | \(e\left(\frac{21}{164}\right)\) | \(e\left(\frac{53}{164}\right)\) | \(e\left(\frac{81}{164}\right)\) | \(e\left(\frac{1}{82}\right)\) |
| \(\chi_{6724}(1157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{164}\right)\) | \(e\left(\frac{1}{82}\right)\) | \(e\left(\frac{17}{164}\right)\) | \(e\left(\frac{3}{82}\right)\) | \(e\left(\frac{53}{164}\right)\) | \(e\left(\frac{97}{164}\right)\) | \(e\left(\frac{87}{164}\right)\) | \(e\left(\frac{79}{164}\right)\) | \(e\left(\frac{31}{164}\right)\) | \(e\left(\frac{51}{82}\right)\) |
| \(\chi_{6724}(1221,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{164}\right)\) | \(e\left(\frac{67}{82}\right)\) | \(e\left(\frac{155}{164}\right)\) | \(e\left(\frac{37}{82}\right)\) | \(e\left(\frac{107}{164}\right)\) | \(e\left(\frac{103}{164}\right)\) | \(e\left(\frac{89}{164}\right)\) | \(e\left(\frac{45}{164}\right)\) | \(e\left(\frac{109}{164}\right)\) | \(e\left(\frac{55}{82}\right)\) |
| \(\chi_{6724}(1321,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{164}\right)\) | \(e\left(\frac{53}{82}\right)\) | \(e\left(\frac{81}{164}\right)\) | \(e\left(\frac{77}{82}\right)\) | \(e\left(\frac{21}{164}\right)\) | \(e\left(\frac{57}{164}\right)\) | \(e\left(\frac{19}{164}\right)\) | \(e\left(\frac{87}{164}\right)\) | \(e\left(\frac{3}{164}\right)\) | \(e\left(\frac{79}{82}\right)\) |
| \(\chi_{6724}(1385,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{164}\right)\) | \(e\left(\frac{15}{82}\right)\) | \(e\left(\frac{91}{164}\right)\) | \(e\left(\frac{45}{82}\right)\) | \(e\left(\frac{139}{164}\right)\) | \(e\left(\frac{143}{164}\right)\) | \(e\left(\frac{157}{164}\right)\) | \(e\left(\frac{37}{164}\right)\) | \(e\left(\frac{137}{164}\right)\) | \(e\left(\frac{27}{82}\right)\) |
| \(\chi_{6724}(1485,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{164}\right)\) | \(e\left(\frac{23}{82}\right)\) | \(e\left(\frac{145}{164}\right)\) | \(e\left(\frac{69}{82}\right)\) | \(e\left(\frac{153}{164}\right)\) | \(e\left(\frac{17}{164}\right)\) | \(e\left(\frac{115}{164}\right)\) | \(e\left(\frac{95}{164}\right)\) | \(e\left(\frac{139}{164}\right)\) | \(e\left(\frac{25}{82}\right)\) |
| \(\chi_{6724}(1549,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{135}{164}\right)\) | \(e\left(\frac{45}{82}\right)\) | \(e\left(\frac{27}{164}\right)\) | \(e\left(\frac{53}{82}\right)\) | \(e\left(\frac{7}{164}\right)\) | \(e\left(\frac{19}{164}\right)\) | \(e\left(\frac{61}{164}\right)\) | \(e\left(\frac{29}{164}\right)\) | \(e\left(\frac{1}{164}\right)\) | \(e\left(\frac{81}{82}\right)\) |
| \(\chi_{6724}(1649,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{164}\right)\) | \(e\left(\frac{75}{82}\right)\) | \(e\left(\frac{45}{164}\right)\) | \(e\left(\frac{61}{82}\right)\) | \(e\left(\frac{121}{164}\right)\) | \(e\left(\frac{141}{164}\right)\) | \(e\left(\frac{47}{164}\right)\) | \(e\left(\frac{103}{164}\right)\) | \(e\left(\frac{111}{164}\right)\) | \(e\left(\frac{53}{82}\right)\) |
| \(\chi_{6724}(1713,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{164}\right)\) | \(e\left(\frac{75}{82}\right)\) | \(e\left(\frac{127}{164}\right)\) | \(e\left(\frac{61}{82}\right)\) | \(e\left(\frac{39}{164}\right)\) | \(e\left(\frac{59}{164}\right)\) | \(e\left(\frac{129}{164}\right)\) | \(e\left(\frac{21}{164}\right)\) | \(e\left(\frac{29}{164}\right)\) | \(e\left(\frac{53}{82}\right)\) |
| \(\chi_{6724}(1813,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{164}\right)\) | \(e\left(\frac{45}{82}\right)\) | \(e\left(\frac{109}{164}\right)\) | \(e\left(\frac{53}{82}\right)\) | \(e\left(\frac{89}{164}\right)\) | \(e\left(\frac{101}{164}\right)\) | \(e\left(\frac{143}{164}\right)\) | \(e\left(\frac{111}{164}\right)\) | \(e\left(\frac{83}{164}\right)\) | \(e\left(\frac{81}{82}\right)\) |
| \(\chi_{6724}(1877,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{164}\right)\) | \(e\left(\frac{23}{82}\right)\) | \(e\left(\frac{63}{164}\right)\) | \(e\left(\frac{69}{82}\right)\) | \(e\left(\frac{71}{164}\right)\) | \(e\left(\frac{99}{164}\right)\) | \(e\left(\frac{33}{164}\right)\) | \(e\left(\frac{13}{164}\right)\) | \(e\left(\frac{57}{164}\right)\) | \(e\left(\frac{25}{82}\right)\) |
| \(\chi_{6724}(1977,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{164}\right)\) | \(e\left(\frac{15}{82}\right)\) | \(e\left(\frac{9}{164}\right)\) | \(e\left(\frac{45}{82}\right)\) | \(e\left(\frac{57}{164}\right)\) | \(e\left(\frac{61}{164}\right)\) | \(e\left(\frac{75}{164}\right)\) | \(e\left(\frac{119}{164}\right)\) | \(e\left(\frac{55}{164}\right)\) | \(e\left(\frac{27}{82}\right)\) |
| \(\chi_{6724}(2041,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{159}{164}\right)\) | \(e\left(\frac{53}{82}\right)\) | \(e\left(\frac{163}{164}\right)\) | \(e\left(\frac{77}{82}\right)\) | \(e\left(\frac{103}{164}\right)\) | \(e\left(\frac{139}{164}\right)\) | \(e\left(\frac{101}{164}\right)\) | \(e\left(\frac{5}{164}\right)\) | \(e\left(\frac{85}{164}\right)\) | \(e\left(\frac{79}{82}\right)\) |
| \(\chi_{6724}(2141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{164}\right)\) | \(e\left(\frac{67}{82}\right)\) | \(e\left(\frac{73}{164}\right)\) | \(e\left(\frac{37}{82}\right)\) | \(e\left(\frac{25}{164}\right)\) | \(e\left(\frac{21}{164}\right)\) | \(e\left(\frac{7}{164}\right)\) | \(e\left(\frac{127}{164}\right)\) | \(e\left(\frac{27}{164}\right)\) | \(e\left(\frac{55}{82}\right)\) |
| \(\chi_{6724}(2205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{164}\right)\) | \(e\left(\frac{1}{82}\right)\) | \(e\left(\frac{99}{164}\right)\) | \(e\left(\frac{3}{82}\right)\) | \(e\left(\frac{135}{164}\right)\) | \(e\left(\frac{15}{164}\right)\) | \(e\left(\frac{5}{164}\right)\) | \(e\left(\frac{161}{164}\right)\) | \(e\left(\frac{113}{164}\right)\) | \(e\left(\frac{51}{82}\right)\) |
| \(\chi_{6724}(2305,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{164}\right)\) | \(e\left(\frac{37}{82}\right)\) | \(e\left(\frac{137}{164}\right)\) | \(e\left(\frac{29}{82}\right)\) | \(e\left(\frac{157}{164}\right)\) | \(e\left(\frac{145}{164}\right)\) | \(e\left(\frac{103}{164}\right)\) | \(e\left(\frac{135}{164}\right)\) | \(e\left(\frac{163}{164}\right)\) | \(e\left(\frac{1}{82}\right)\) |
| \(\chi_{6724}(2369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{164}\right)\) | \(e\left(\frac{31}{82}\right)\) | \(e\left(\frac{35}{164}\right)\) | \(e\left(\frac{11}{82}\right)\) | \(e\left(\frac{3}{164}\right)\) | \(e\left(\frac{55}{164}\right)\) | \(e\left(\frac{73}{164}\right)\) | \(e\left(\frac{153}{164}\right)\) | \(e\left(\frac{141}{164}\right)\) | \(e\left(\frac{23}{82}\right)\) |
| \(\chi_{6724}(2469,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{164}\right)\) | \(e\left(\frac{7}{82}\right)\) | \(e\left(\frac{37}{164}\right)\) | \(e\left(\frac{21}{82}\right)\) | \(e\left(\frac{125}{164}\right)\) | \(e\left(\frac{105}{164}\right)\) | \(e\left(\frac{35}{164}\right)\) | \(e\left(\frac{143}{164}\right)\) | \(e\left(\frac{135}{164}\right)\) | \(e\left(\frac{29}{82}\right)\) |