from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(289, base_ring=CyclotomicField(272))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,289))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(289\) | |
| Conductor: | \(289\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(272\) | 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: | $\Q(\zeta_{272})$ |
| Fixed field: | Number field defined by a degree 272 polynomial (not computed) |
First 31 of 128 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{289}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{136}\right)\) | \(e\left(\frac{1}{272}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{229}{272}\right)\) | \(e\left(\frac{191}{272}\right)\) | \(e\left(\frac{155}{272}\right)\) | \(e\left(\frac{13}{136}\right)\) | \(e\left(\frac{1}{136}\right)\) | \(e\left(\frac{147}{272}\right)\) | \(e\left(\frac{23}{272}\right)\) |
| \(\chi_{289}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{136}\right)\) | \(e\left(\frac{229}{272}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{217}{272}\right)\) | \(e\left(\frac{219}{272}\right)\) | \(e\left(\frac{135}{272}\right)\) | \(e\left(\frac{121}{136}\right)\) | \(e\left(\frac{93}{136}\right)\) | \(e\left(\frac{207}{272}\right)\) | \(e\left(\frac{99}{272}\right)\) |
| \(\chi_{289}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{136}\right)\) | \(e\left(\frac{191}{272}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{219}{272}\right)\) | \(e\left(\frac{33}{272}\right)\) | \(e\left(\frac{229}{272}\right)\) | \(e\left(\frac{35}{136}\right)\) | \(e\left(\frac{55}{136}\right)\) | \(e\left(\frac{61}{272}\right)\) | \(e\left(\frac{41}{272}\right)\) |
| \(\chi_{289}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{136}\right)\) | \(e\left(\frac{155}{272}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{135}{272}\right)\) | \(e\left(\frac{229}{272}\right)\) | \(e\left(\frac{89}{272}\right)\) | \(e\left(\frac{111}{136}\right)\) | \(e\left(\frac{19}{136}\right)\) | \(e\left(\frac{209}{272}\right)\) | \(e\left(\frac{29}{272}\right)\) |
| \(\chi_{289}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{93}{136}\right)\) | \(e\left(\frac{147}{272}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{207}{272}\right)\) | \(e\left(\frac{61}{272}\right)\) | \(e\left(\frac{209}{272}\right)\) | \(e\left(\frac{7}{136}\right)\) | \(e\left(\frac{11}{136}\right)\) | \(e\left(\frac{121}{272}\right)\) | \(e\left(\frac{117}{272}\right)\) |
| \(\chi_{289}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{136}\right)\) | \(e\left(\frac{23}{272}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{99}{272}\right)\) | \(e\left(\frac{41}{272}\right)\) | \(e\left(\frac{29}{272}\right)\) | \(e\left(\frac{27}{136}\right)\) | \(e\left(\frac{23}{136}\right)\) | \(e\left(\frac{117}{272}\right)\) | \(e\left(\frac{257}{272}\right)\) |
| \(\chi_{289}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{136}\right)\) | \(e\left(\frac{109}{272}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{209}{272}\right)\) | \(e\left(\frac{147}{272}\right)\) | \(e\left(\frac{31}{272}\right)\) | \(e\left(\frac{57}{136}\right)\) | \(e\left(\frac{109}{136}\right)\) | \(e\left(\frac{247}{272}\right)\) | \(e\left(\frac{59}{272}\right)\) |
| \(\chi_{289}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{135}{136}\right)\) | \(e\left(\frac{73}{272}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{125}{272}\right)\) | \(e\left(\frac{71}{272}\right)\) | \(e\left(\frac{163}{272}\right)\) | \(e\left(\frac{133}{136}\right)\) | \(e\left(\frac{73}{136}\right)\) | \(e\left(\frac{123}{272}\right)\) | \(e\left(\frac{47}{272}\right)\) |
| \(\chi_{289}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{136}\right)\) | \(e\left(\frac{65}{272}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{197}{272}\right)\) | \(e\left(\frac{175}{272}\right)\) | \(e\left(\frac{11}{272}\right)\) | \(e\left(\frac{29}{136}\right)\) | \(e\left(\frac{65}{136}\right)\) | \(e\left(\frac{35}{272}\right)\) | \(e\left(\frac{135}{272}\right)\) |
| \(\chi_{289}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{136}\right)\) | \(e\left(\frac{213}{272}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{89}{272}\right)\) | \(e\left(\frac{155}{272}\right)\) | \(e\left(\frac{103}{272}\right)\) | \(e\left(\frac{49}{136}\right)\) | \(e\left(\frac{77}{136}\right)\) | \(e\left(\frac{31}{272}\right)\) | \(e\left(\frac{3}{272}\right)\) |
| \(\chi_{289}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{105}{136}\right)\) | \(e\left(\frac{223}{272}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{203}{272}\right)\) | \(e\left(\frac{161}{272}\right)\) | \(e\left(\frac{21}{272}\right)\) | \(e\left(\frac{43}{136}\right)\) | \(e\left(\frac{87}{136}\right)\) | \(e\left(\frac{141}{272}\right)\) | \(e\left(\frac{233}{272}\right)\) |
| \(\chi_{289}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{117}{136}\right)\) | \(e\left(\frac{27}{272}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{199}{272}\right)\) | \(e\left(\frac{261}{272}\right)\) | \(e\left(\frac{105}{272}\right)\) | \(e\left(\frac{79}{136}\right)\) | \(e\left(\frac{27}{136}\right)\) | \(e\left(\frac{161}{272}\right)\) | \(e\left(\frac{77}{272}\right)\) |
| \(\chi_{289}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{136}\right)\) | \(e\left(\frac{3}{272}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{143}{272}\right)\) | \(e\left(\frac{29}{272}\right)\) | \(e\left(\frac{193}{272}\right)\) | \(e\left(\frac{39}{136}\right)\) | \(e\left(\frac{3}{136}\right)\) | \(e\left(\frac{169}{272}\right)\) | \(e\left(\frac{69}{272}\right)\) |
| \(\chi_{289}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{136}\right)\) | \(e\left(\frac{263}{272}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{115}{272}\right)\) | \(e\left(\frac{185}{272}\right)\) | \(e\left(\frac{237}{272}\right)\) | \(e\left(\frac{19}{136}\right)\) | \(e\left(\frac{127}{136}\right)\) | \(e\left(\frac{37}{272}\right)\) | \(e\left(\frac{65}{272}\right)\) |
| \(\chi_{289}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{136}\right)\) | \(e\left(\frac{125}{272}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{65}{272}\right)\) | \(e\left(\frac{211}{272}\right)\) | \(e\left(\frac{63}{272}\right)\) | \(e\left(\frac{129}{136}\right)\) | \(e\left(\frac{125}{136}\right)\) | \(e\left(\frac{151}{272}\right)\) | \(e\left(\frac{155}{272}\right)\) |
| \(\chi_{289}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{136}\right)\) | \(e\left(\frac{9}{272}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{157}{272}\right)\) | \(e\left(\frac{87}{272}\right)\) | \(e\left(\frac{35}{272}\right)\) | \(e\left(\frac{117}{136}\right)\) | \(e\left(\frac{9}{136}\right)\) | \(e\left(\frac{235}{272}\right)\) | \(e\left(\frac{207}{272}\right)\) |
| \(\chi_{289}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{136}\right)\) | \(e\left(\frac{129}{272}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{165}{272}\right)\) | \(e\left(\frac{159}{272}\right)\) | \(e\left(\frac{139}{272}\right)\) | \(e\left(\frac{45}{136}\right)\) | \(e\left(\frac{129}{136}\right)\) | \(e\left(\frac{195}{272}\right)\) | \(e\left(\frac{247}{272}\right)\) |
| \(\chi_{289}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{136}\right)\) | \(e\left(\frac{197}{272}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{233}{272}\right)\) | \(e\left(\frac{91}{272}\right)\) | \(e\left(\frac{71}{272}\right)\) | \(e\left(\frac{113}{136}\right)\) | \(e\left(\frac{61}{136}\right)\) | \(e\left(\frac{127}{272}\right)\) | \(e\left(\frac{179}{272}\right)\) |
| \(\chi_{289}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{136}\right)\) | \(e\left(\frac{171}{272}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{263}{272}\right)\) | \(e\left(\frac{21}{272}\right)\) | \(e\left(\frac{121}{272}\right)\) | \(e\left(\frac{47}{136}\right)\) | \(e\left(\frac{35}{136}\right)\) | \(e\left(\frac{113}{272}\right)\) | \(e\left(\frac{125}{272}\right)\) |
| \(\chi_{289}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{136}\right)\) | \(e\left(\frac{131}{272}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{79}{272}\right)\) | \(e\left(\frac{269}{272}\right)\) | \(e\left(\frac{177}{272}\right)\) | \(e\left(\frac{71}{136}\right)\) | \(e\left(\frac{131}{136}\right)\) | \(e\left(\frac{217}{272}\right)\) | \(e\left(\frac{21}{272}\right)\) |
| \(\chi_{289}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{136}\right)\) | \(e\left(\frac{231}{272}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{131}{272}\right)\) | \(e\left(\frac{57}{272}\right)\) | \(e\left(\frac{173}{272}\right)\) | \(e\left(\frac{11}{136}\right)\) | \(e\left(\frac{95}{136}\right)\) | \(e\left(\frac{229}{272}\right)\) | \(e\left(\frac{145}{272}\right)\) |
| \(\chi_{289}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{136}\right)\) | \(e\left(\frac{141}{272}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{193}{272}\right)\) | \(e\left(\frac{3}{272}\right)\) | \(e\left(\frac{95}{272}\right)\) | \(e\left(\frac{65}{136}\right)\) | \(e\left(\frac{5}{136}\right)\) | \(e\left(\frac{55}{272}\right)\) | \(e\left(\frac{251}{272}\right)\) |
| \(\chi_{289}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{136}\right)\) | \(e\left(\frac{217}{272}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{189}{272}\right)\) | \(e\left(\frac{103}{272}\right)\) | \(e\left(\frac{179}{272}\right)\) | \(e\left(\frac{101}{136}\right)\) | \(e\left(\frac{81}{136}\right)\) | \(e\left(\frac{75}{272}\right)\) | \(e\left(\frac{95}{272}\right)\) |
| \(\chi_{289}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{111}{136}\right)\) | \(e\left(\frac{193}{272}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{133}{272}\right)\) | \(e\left(\frac{143}{272}\right)\) | \(e\left(\frac{267}{272}\right)\) | \(e\left(\frac{61}{136}\right)\) | \(e\left(\frac{57}{136}\right)\) | \(e\left(\frac{83}{272}\right)\) | \(e\left(\frac{87}{272}\right)\) |
| \(\chi_{289}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{136}\right)\) | \(e\left(\frac{181}{272}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{105}{272}\right)\) | \(e\left(\frac{27}{272}\right)\) | \(e\left(\frac{39}{272}\right)\) | \(e\left(\frac{41}{136}\right)\) | \(e\left(\frac{45}{136}\right)\) | \(e\left(\frac{223}{272}\right)\) | \(e\left(\frac{83}{272}\right)\) |
| \(\chi_{289}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{136}\right)\) | \(e\left(\frac{15}{272}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{171}{272}\right)\) | \(e\left(\frac{145}{272}\right)\) | \(e\left(\frac{149}{272}\right)\) | \(e\left(\frac{59}{136}\right)\) | \(e\left(\frac{15}{136}\right)\) | \(e\left(\frac{29}{272}\right)\) | \(e\left(\frac{73}{272}\right)\) |
| \(\chi_{289}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{136}\right)\) | \(e\left(\frac{43}{272}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{55}{272}\right)\) | \(e\left(\frac{53}{272}\right)\) | \(e\left(\frac{137}{272}\right)\) | \(e\left(\frac{15}{136}\right)\) | \(e\left(\frac{43}{136}\right)\) | \(e\left(\frac{65}{272}\right)\) | \(e\left(\frac{173}{272}\right)\) |
| \(\chi_{289}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{136}\right)\) | \(e\left(\frac{259}{272}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{15}{272}\right)\) | \(e\left(\frac{237}{272}\right)\) | \(e\left(\frac{161}{272}\right)\) | \(e\left(\frac{103}{136}\right)\) | \(e\left(\frac{123}{136}\right)\) | \(e\left(\frac{265}{272}\right)\) | \(e\left(\frac{245}{272}\right)\) |
| \(\chi_{289}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{136}\right)\) | \(e\left(\frac{199}{272}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{147}{272}\right)\) | \(e\left(\frac{201}{272}\right)\) | \(e\left(\frac{109}{272}\right)\) | \(e\left(\frac{3}{136}\right)\) | \(e\left(\frac{63}{136}\right)\) | \(e\left(\frac{149}{272}\right)\) | \(e\left(\frac{225}{272}\right)\) |
| \(\chi_{289}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{136}\right)\) | \(e\left(\frac{157}{272}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{49}{272}\right)\) | \(e\left(\frac{67}{272}\right)\) | \(e\left(\frac{127}{272}\right)\) | \(e\left(\frac{1}{136}\right)\) | \(e\left(\frac{21}{136}\right)\) | \(e\left(\frac{231}{272}\right)\) | \(e\left(\frac{75}{272}\right)\) |
| \(\chi_{289}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{136}\right)\) | \(e\left(\frac{257}{272}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{101}{272}\right)\) | \(e\left(\frac{127}{272}\right)\) | \(e\left(\frac{123}{272}\right)\) | \(e\left(\frac{77}{136}\right)\) | \(e\left(\frac{121}{136}\right)\) | \(e\left(\frac{243}{272}\right)\) | \(e\left(\frac{199}{272}\right)\) |