from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(647, base_ring=CyclotomicField(646))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,647))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(647\) | |
| Conductor: | \(647\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(646\) | 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: | $\Q(\zeta_{323})$ |
| Fixed field: | Number field defined by a degree 646 polynomial (not computed) |
First 31 of 288 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{647}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{323}\right)\) | \(e\left(\frac{183}{323}\right)\) | \(e\left(\frac{158}{323}\right)\) | \(e\left(\frac{1}{646}\right)\) | \(e\left(\frac{262}{323}\right)\) | \(e\left(\frac{69}{323}\right)\) | \(e\left(\frac{237}{323}\right)\) | \(e\left(\frac{43}{323}\right)\) | \(e\left(\frac{159}{646}\right)\) | \(e\left(\frac{169}{646}\right)\) |
| \(\chi_{647}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{287}{323}\right)\) | \(e\left(\frac{27}{323}\right)\) | \(e\left(\frac{251}{323}\right)\) | \(e\left(\frac{159}{646}\right)\) | \(e\left(\frac{314}{323}\right)\) | \(e\left(\frac{312}{323}\right)\) | \(e\left(\frac{215}{323}\right)\) | \(e\left(\frac{54}{323}\right)\) | \(e\left(\frac{87}{646}\right)\) | \(e\left(\frac{385}{646}\right)\) |
| \(\chi_{647}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{108}{323}\right)\) | \(e\left(\frac{242}{323}\right)\) | \(e\left(\frac{216}{323}\right)\) | \(e\left(\frac{169}{646}\right)\) | \(e\left(\frac{27}{323}\right)\) | \(e\left(\frac{33}{323}\right)\) | \(e\left(\frac{1}{323}\right)\) | \(e\left(\frac{161}{323}\right)\) | \(e\left(\frac{385}{646}\right)\) | \(e\left(\frac{137}{646}\right)\) |
| \(\chi_{647}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{246}{323}\right)\) | \(e\left(\frac{300}{323}\right)\) | \(e\left(\frac{169}{323}\right)\) | \(e\left(\frac{367}{646}\right)\) | \(e\left(\frac{223}{323}\right)\) | \(e\left(\frac{129}{323}\right)\) | \(e\left(\frac{92}{323}\right)\) | \(e\left(\frac{277}{323}\right)\) | \(e\left(\frac{213}{646}\right)\) | \(e\left(\frac{7}{646}\right)\) |
| \(\chi_{647}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{220}{323}\right)\) | \(e\left(\frac{158}{323}\right)\) | \(e\left(\frac{117}{323}\right)\) | \(e\left(\frac{105}{646}\right)\) | \(e\left(\frac{55}{323}\right)\) | \(e\left(\frac{139}{323}\right)\) | \(e\left(\frac{14}{323}\right)\) | \(e\left(\frac{316}{323}\right)\) | \(e\left(\frac{545}{646}\right)\) | \(e\left(\frac{303}{646}\right)\) |
| \(\chi_{647}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{172}{323}\right)\) | \(e\left(\frac{194}{323}\right)\) | \(e\left(\frac{21}{323}\right)\) | \(e\left(\frac{317}{646}\right)\) | \(e\left(\frac{43}{323}\right)\) | \(e\left(\frac{232}{323}\right)\) | \(e\left(\frac{193}{323}\right)\) | \(e\left(\frac{65}{323}\right)\) | \(e\left(\frac{15}{646}\right)\) | \(e\left(\frac{601}{646}\right)\) |
| \(\chi_{647}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{316}{323}\right)\) | \(e\left(\frac{86}{323}\right)\) | \(e\left(\frac{309}{323}\right)\) | \(e\left(\frac{327}{646}\right)\) | \(e\left(\frac{79}{323}\right)\) | \(e\left(\frac{276}{323}\right)\) | \(e\left(\frac{302}{323}\right)\) | \(e\left(\frac{172}{323}\right)\) | \(e\left(\frac{313}{646}\right)\) | \(e\left(\frac{353}{646}\right)\) |
| \(\chi_{647}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{154}{323}\right)\) | \(e\left(\frac{46}{323}\right)\) | \(e\left(\frac{308}{323}\right)\) | \(e\left(\frac{235}{646}\right)\) | \(e\left(\frac{200}{323}\right)\) | \(e\left(\frac{65}{323}\right)\) | \(e\left(\frac{139}{323}\right)\) | \(e\left(\frac{92}{323}\right)\) | \(e\left(\frac{543}{646}\right)\) | \(e\left(\frac{309}{646}\right)\) |
| \(\chi_{647}(30,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{323}\right)\) | \(e\left(\frac{144}{323}\right)\) | \(e\left(\frac{262}{323}\right)\) | \(e\left(\frac{525}{646}\right)\) | \(e\left(\frac{275}{323}\right)\) | \(e\left(\frac{49}{323}\right)\) | \(e\left(\frac{70}{323}\right)\) | \(e\left(\frac{288}{323}\right)\) | \(e\left(\frac{141}{646}\right)\) | \(e\left(\frac{223}{646}\right)\) |
| \(\chi_{647}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{275}{323}\right)\) | \(e\left(\frac{36}{323}\right)\) | \(e\left(\frac{227}{323}\right)\) | \(e\left(\frac{535}{646}\right)\) | \(e\left(\frac{311}{323}\right)\) | \(e\left(\frac{93}{323}\right)\) | \(e\left(\frac{179}{323}\right)\) | \(e\left(\frac{72}{323}\right)\) | \(e\left(\frac{439}{646}\right)\) | \(e\left(\frac{621}{646}\right)\) |
| \(\chi_{647}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{322}{323}\right)\) | \(e\left(\frac{243}{323}\right)\) | \(e\left(\frac{321}{323}\right)\) | \(e\left(\frac{139}{646}\right)\) | \(e\left(\frac{242}{323}\right)\) | \(e\left(\frac{224}{323}\right)\) | \(e\left(\frac{320}{323}\right)\) | \(e\left(\frac{163}{323}\right)\) | \(e\left(\frac{137}{646}\right)\) | \(e\left(\frac{235}{646}\right)\) |
| \(\chi_{647}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{323}\right)\) | \(e\left(\frac{80}{323}\right)\) | \(e\left(\frac{2}{323}\right)\) | \(e\left(\frac{507}{646}\right)\) | \(e\left(\frac{81}{323}\right)\) | \(e\left(\frac{99}{323}\right)\) | \(e\left(\frac{3}{323}\right)\) | \(e\left(\frac{160}{323}\right)\) | \(e\left(\frac{509}{646}\right)\) | \(e\left(\frac{411}{646}\right)\) |
| \(\chi_{647}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{105}{323}\right)\) | \(e\left(\frac{2}{323}\right)\) | \(e\left(\frac{210}{323}\right)\) | \(e\left(\frac{263}{646}\right)\) | \(e\left(\frac{107}{323}\right)\) | \(e\left(\frac{59}{323}\right)\) | \(e\left(\frac{315}{323}\right)\) | \(e\left(\frac{4}{323}\right)\) | \(e\left(\frac{473}{646}\right)\) | \(e\left(\frac{519}{646}\right)\) |
| \(\chi_{647}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{201}{323}\right)\) | \(e\left(\frac{253}{323}\right)\) | \(e\left(\frac{79}{323}\right)\) | \(e\left(\frac{485}{646}\right)\) | \(e\left(\frac{131}{323}\right)\) | \(e\left(\frac{196}{323}\right)\) | \(e\left(\frac{280}{323}\right)\) | \(e\left(\frac{183}{323}\right)\) | \(e\left(\frac{241}{646}\right)\) | \(e\left(\frac{569}{646}\right)\) |
| \(\chi_{647}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{90}{323}\right)\) | \(e\left(\frac{94}{323}\right)\) | \(e\left(\frac{180}{323}\right)\) | \(e\left(\frac{87}{646}\right)\) | \(e\left(\frac{184}{323}\right)\) | \(e\left(\frac{189}{323}\right)\) | \(e\left(\frac{270}{323}\right)\) | \(e\left(\frac{188}{323}\right)\) | \(e\left(\frac{267}{646}\right)\) | \(e\left(\frac{491}{646}\right)\) |
| \(\chi_{647}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{323}\right)\) | \(e\left(\frac{213}{323}\right)\) | \(e\left(\frac{78}{323}\right)\) | \(e\left(\frac{393}{646}\right)\) | \(e\left(\frac{252}{323}\right)\) | \(e\left(\frac{308}{323}\right)\) | \(e\left(\frac{117}{323}\right)\) | \(e\left(\frac{103}{323}\right)\) | \(e\left(\frac{471}{646}\right)\) | \(e\left(\frac{525}{646}\right)\) |
| \(\chi_{647}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{303}{323}\right)\) | \(e\left(\frac{15}{323}\right)\) | \(e\left(\frac{283}{323}\right)\) | \(e\left(\frac{519}{646}\right)\) | \(e\left(\frac{318}{323}\right)\) | \(e\left(\frac{281}{323}\right)\) | \(e\left(\frac{263}{323}\right)\) | \(e\left(\frac{30}{323}\right)\) | \(e\left(\frac{479}{646}\right)\) | \(e\left(\frac{501}{646}\right)\) |
| \(\chi_{647}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{64}{323}\right)\) | \(e\left(\frac{275}{323}\right)\) | \(e\left(\frac{128}{323}\right)\) | \(e\left(\frac{471}{646}\right)\) | \(e\left(\frac{16}{323}\right)\) | \(e\left(\frac{199}{323}\right)\) | \(e\left(\frac{192}{323}\right)\) | \(e\left(\frac{227}{323}\right)\) | \(e\left(\frac{599}{646}\right)\) | \(e\left(\frac{141}{646}\right)\) |
| \(\chi_{647}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{100}{323}\right)\) | \(e\left(\frac{248}{323}\right)\) | \(e\left(\frac{200}{323}\right)\) | \(e\left(\frac{635}{646}\right)\) | \(e\left(\frac{25}{323}\right)\) | \(e\left(\frac{210}{323}\right)\) | \(e\left(\frac{300}{323}\right)\) | \(e\left(\frac{173}{323}\right)\) | \(e\left(\frac{189}{646}\right)\) | \(e\left(\frac{79}{646}\right)\) |
| \(\chi_{647}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{323}\right)\) | \(e\left(\frac{311}{323}\right)\) | \(e\left(\frac{32}{323}\right)\) | \(e\left(\frac{37}{646}\right)\) | \(e\left(\frac{4}{323}\right)\) | \(e\left(\frac{292}{323}\right)\) | \(e\left(\frac{48}{323}\right)\) | \(e\left(\frac{299}{323}\right)\) | \(e\left(\frac{69}{646}\right)\) | \(e\left(\frac{439}{646}\right)\) |
| \(\chi_{647}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{323}\right)\) | \(e\left(\frac{53}{323}\right)\) | \(e\left(\frac{74}{323}\right)\) | \(e\left(\frac{25}{646}\right)\) | \(e\left(\frac{90}{323}\right)\) | \(e\left(\frac{110}{323}\right)\) | \(e\left(\frac{111}{323}\right)\) | \(e\left(\frac{106}{323}\right)\) | \(e\left(\frac{99}{646}\right)\) | \(e\left(\frac{349}{646}\right)\) |
| \(\chi_{647}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{203}{323}\right)\) | \(e\left(\frac{90}{323}\right)\) | \(e\left(\frac{83}{323}\right)\) | \(e\left(\frac{207}{646}\right)\) | \(e\left(\frac{293}{323}\right)\) | \(e\left(\frac{71}{323}\right)\) | \(e\left(\frac{286}{323}\right)\) | \(e\left(\frac{180}{323}\right)\) | \(e\left(\frac{613}{646}\right)\) | \(e\left(\frac{99}{646}\right)\) |
| \(\chi_{647}(66,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{160}{323}\right)\) | \(e\left(\frac{203}{323}\right)\) | \(e\left(\frac{320}{323}\right)\) | \(e\left(\frac{47}{646}\right)\) | \(e\left(\frac{40}{323}\right)\) | \(e\left(\frac{13}{323}\right)\) | \(e\left(\frac{157}{323}\right)\) | \(e\left(\frac{83}{323}\right)\) | \(e\left(\frac{367}{646}\right)\) | \(e\left(\frac{191}{646}\right)\) |
| \(\chi_{647}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{321}{323}\right)\) | \(e\left(\frac{163}{323}\right)\) | \(e\left(\frac{319}{323}\right)\) | \(e\left(\frac{601}{646}\right)\) | \(e\left(\frac{161}{323}\right)\) | \(e\left(\frac{125}{323}\right)\) | \(e\left(\frac{317}{323}\right)\) | \(e\left(\frac{3}{323}\right)\) | \(e\left(\frac{597}{646}\right)\) | \(e\left(\frac{147}{646}\right)\) |
| \(\chi_{647}(70,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{207}{323}\right)\) | \(e\left(\frac{87}{323}\right)\) | \(e\left(\frac{91}{323}\right)\) | \(e\left(\frac{297}{646}\right)\) | \(e\left(\frac{294}{323}\right)\) | \(e\left(\frac{144}{323}\right)\) | \(e\left(\frac{298}{323}\right)\) | \(e\left(\frac{174}{323}\right)\) | \(e\left(\frac{65}{646}\right)\) | \(e\left(\frac{451}{646}\right)\) |
| \(\chi_{647}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{290}{323}\right)\) | \(e\left(\frac{267}{323}\right)\) | \(e\left(\frac{257}{323}\right)\) | \(e\left(\frac{65}{646}\right)\) | \(e\left(\frac{234}{323}\right)\) | \(e\left(\frac{286}{323}\right)\) | \(e\left(\frac{224}{323}\right)\) | \(e\left(\frac{211}{323}\right)\) | \(e\left(\frac{645}{646}\right)\) | \(e\left(\frac{3}{646}\right)\) |
| \(\chi_{647}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{276}{323}\right)\) | \(e\left(\frac{116}{323}\right)\) | \(e\left(\frac{229}{323}\right)\) | \(e\left(\frac{73}{646}\right)\) | \(e\left(\frac{69}{323}\right)\) | \(e\left(\frac{192}{323}\right)\) | \(e\left(\frac{182}{323}\right)\) | \(e\left(\frac{232}{323}\right)\) | \(e\left(\frac{625}{646}\right)\) | \(e\left(\frac{63}{646}\right)\) |
| \(\chi_{647}(76,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{313}{323}\right)\) | \(e\left(\frac{169}{323}\right)\) | \(e\left(\frac{303}{323}\right)\) | \(e\left(\frac{421}{646}\right)\) | \(e\left(\frac{159}{323}\right)\) | \(e\left(\frac{302}{323}\right)\) | \(e\left(\frac{293}{323}\right)\) | \(e\left(\frac{15}{323}\right)\) | \(e\left(\frac{401}{646}\right)\) | \(e\left(\frac{89}{646}\right)\) |
| \(\chi_{647}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{28}{323}\right)\) | \(e\left(\frac{302}{323}\right)\) | \(e\left(\frac{56}{323}\right)\) | \(e\left(\frac{307}{646}\right)\) | \(e\left(\frac{7}{323}\right)\) | \(e\left(\frac{188}{323}\right)\) | \(e\left(\frac{84}{323}\right)\) | \(e\left(\frac{281}{323}\right)\) | \(e\left(\frac{363}{646}\right)\) | \(e\left(\frac{203}{646}\right)\) |
| \(\chi_{647}(80,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{265}{323}\right)\) | \(e\left(\frac{205}{323}\right)\) | \(e\left(\frac{207}{323}\right)\) | \(e\left(\frac{633}{646}\right)\) | \(e\left(\frac{147}{323}\right)\) | \(e\left(\frac{72}{323}\right)\) | \(e\left(\frac{149}{323}\right)\) | \(e\left(\frac{87}{323}\right)\) | \(e\left(\frac{517}{646}\right)\) | \(e\left(\frac{387}{646}\right)\) |
| \(\chi_{647}(85,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{297}{323}\right)\) | \(e\left(\frac{181}{323}\right)\) | \(e\left(\frac{271}{323}\right)\) | \(e\left(\frac{61}{646}\right)\) | \(e\left(\frac{155}{323}\right)\) | \(e\left(\frac{10}{323}\right)\) | \(e\left(\frac{245}{323}\right)\) | \(e\left(\frac{39}{323}\right)\) | \(e\left(\frac{9}{646}\right)\) | \(e\left(\frac{619}{646}\right)\) |